# Combined analysis of neutrino decoherence at reactor experiments

André de Gouvêa<EMAIL_ADDRESS>Northwestern University, Department
of Physics & Astronomy, 2145 Sheridan Road, Evanston, IL 60208, USA  Valentina
De Romeri<EMAIL_ADDRESS>Institut de Física Corpuscular CSIC/Universitat
de València, Parc Científic de Paterna
C/ Catedrático José Beltrán, 2 E-46980 Paterna (Valencia) - Spain  Christoph
A. Ternes<EMAIL_ADDRESS>INFN, Sezione di Torino, Via P. Giuria 1, I–10125
Torino, Italy

###### Abstract

Reactor experiments are well suited to probe the possible loss of coherence of
neutrino oscillations due to wave-packets separation. We combine data from the
short-baseline experiments Daya Bay and the Reactor Experiment for Neutrino
Oscillation (RENO) and from the long baseline reactor experiment KamLAND to
obtain the best current limit on the reactor antineutrino wave-packet width,
$\sigma>2.1\times 10^{-4}$ nm at 90% CL. We also find that the determination
of standard oscillation parameters is robust, i.e., it is mostly insensitive
to the presence of hypothetical decoherence effects once one combines the
results of the different reactor neutrino experiments.

neutrino oscillations, reactor experiments, neutrino decoherence

††preprint: NUHEP-TH/21-02

##  I Introduction

Neutrino oscillations are a consequence of nonzero neutrino masses and the
fact that virtually all useful neutrino sources are coherent, i.e., the
neutrinos produced via charged-current weak interactions associated with
$\ell_{\alpha}$ charged leptons, $\alpha=e,\mu,\tau$, can be faithfully
described as coherent superpositions of neutrinos $\nu_{i}$ with different
masses $m_{i}$, $i=1,2,3$, weighted by the elements $U_{\alpha i}$ of the
leptonic mixing matrix. This is true due in no small part to the fact that
neutrino masses are tiny when compared with their laboratory energies.

While no neutrino source is perfectly coherent, in practice, in the
literature, all potential decoherence effects are neglected. This is easy to
understand. Pragmatically, real-world considerations, including the finite-
size of neutrino sources/detectors and the energy resolution of detecting
devices, lead to effects that mimic quantum mechanical decoherence and, for
the most part, are quantitatively overwhelming when compared to semi-realistic
estimates of the expected fundamental-physics effects. Nonetheless,
oscillation experiments can be used to constrain how coherent the different
neutrino sources are.

Nuclear reactors are excellent sources of antineutrinos and reactor
antineutrino experiments are powerful sources of information on the parameters
that describe neutrino oscillations. Today, reactor experiments provide the
most precise information on the mass-squared difference $\Delta
m^{2}_{21}\equiv m_{2}^{2}-m_{1}^{2}$ and the magnitude of the $U_{e3}$
($\sin\theta_{13}$) element of the leptonic mixing matrix***We use the PDG
parameterization for the neutrino oscillation parameters Zyla:2020zbs .. They
also provide, independent from all other “types” of neutrino experiments,
valuable information on the magnitude of $\Delta m^{2}_{31}\equiv
m_{3}^{2}-m_{1}^{2}$ and the magnitude of the product $U_{e1}^{*}U_{e2}$
($\propto\sin 2\theta_{12}$). Reactor experiments are only insensitive to the
so-called atmospheric mixing angle $\theta_{23}$ and the CP-odd phase
$\delta$. In the near future, the Jiangmen Underground Neutrino Observatory
(JUNO) is expected to extract the most precise measurement of
$\sin^{2}2\theta_{12}$ and, perhaps, determine the neutrino mass ordering
(sign of $\Delta m^{2}_{31}$) Abusleme:2021zrw .

In deGouvea:2020hfl , we argued that reactor neutrino experiments can also be
used to place interesting constraints on how coherent nuclear reactors are as
sources of antineutrinos. There, we concentrated on current constraints from
the km-baseline experiments Daya Bay, in China, and the Reactor Experiment for
Neutrino Oscillation (RENO), in South Korea, and the near-future JUNO
experiment. These experiments are characterized, to a very good approximation,
by a single, well-known baseline. In deGouvea:2020hfl , we found, for RENO and
Daya Bay data, nontrivial correlations between $\sin^{2}\theta_{13},|\Delta
m^{2}_{31}|$ and the decoherence parameter $\sigma$. Future JUNO data, in the
absence of strong decoherence effects, are expected to significantly weaken
these correlations.

Here, we extend our previous analyses and include data from the Kamioka Liquid
Scintillator Antineutrino Detector (KamLAND), which ran for over a decade
starting in 2002. Unlike Daya Bay and RENO, KamLAND was sensitive to neutrinos
from a large number of nuclear reactor sites located between, roughly, 100 km
and 1000 km away and was not characterized by a single baseline. While Daya
Bay and RENO are only sensitive to $\sin^{2}\theta_{13}$ and $|\Delta
m^{2}_{31}|$, KamLAND is sensitive, given that $|U_{e3}|^{2}$ is small, only
to $\sin^{2}2\theta_{12}$ and $|\Delta m^{2}_{21}|$ so the data sets, in some
sense, complement one another unobstructively. The hypothesis that all reactor
experiments are characterized by the same decoherence parameter allows the
KamLAND and Daya Bay/RENO data sets to “inform” one another in nontrivial
ways.

We address the following questions: (i) are the existing reactor neutrino
experiments consistent with the hypothesis that nuclear reactors are a source
of perfectly coherent antineutrinos (the answer is ‘yes’)?, (ii) if the
decoherence parameters of all antineutrinos from nuclear reactors are the
same, how well can existing reactor experiments constrain them?, and (iii) if
one allows for nontrivial values of the decoherence parameters, how much is
the measurement of the different oscillation parameters impacted?

The paper is structured as follows. In Sec. II, we discuss neutrino
oscillation probabilities including quantum decoherence and matter effects. In
Sec. III, we detail our combined analysis of the existing data from the
reactor experiments KamLAND, RENO, and Daya Bay. In Sec. IV, we derive the
best current bound on the neutrino wave-packet width and discuss how the
presence of neutrino decoherence may affect the extraction of standard
oscillation parameters. We draw our conclusions in Sec. V.

##  II Neutrino oscillations including decoherence in matter

Reactor neutrino experiments detect the flux of electron antineutrinos
produced in nuclear reactor cores through the inverse beta-decay process,
$\bar{\nu}_{e}+p\rightarrow e^{+}+n$. Keeping in mind uncertainties on the
antineutrino flux produced in the nuclear fission processes, these experiments
can measure the $\bar{\nu}_{e}$ survival probability
$P(\bar{\nu}_{e}\to\bar{\nu}_{e})$, which depends on the energy of the
antineutrinos $E$ and the baseline $L$, the distance between the source and
the detector.

While decoherence effects may stem from several different physical origins
Kiers:1995zj ; Ohlsson:2000mj ; Beuthe:2001rc ; Beuthe:2002ej ; Giunti:2003ax
; Blennow:2005yk ; Farzan:2008eg ; Kayser:2010pr ; Naumov:2010um ;
Naumov:2013uia ; Jones:2014sfa ; Akhmedov:2019iyt ; Grimus:2019hlq ;
Naumov:2020yyv , here we focus on the possible loss of flavor-coherence of the
neutrino beam that grows with the baseline (often referred to as wave-packet
separation) and is parameterized through the damping parameters

$\xi_{jk}(L,E)=\bigg{(}\frac{L}{L^{\rm coh}_{jk}}\bigg{)}^{2}~{}~{}{\rm
with}~{}~{}j,k=1,2,3.$ (1)

We refer to deGouvea:2020hfl  for a more careful discussion of these
parameters and their effects. If no loss of coherence occurs during neutrino
propagation, $\xi_{jk}(=\xi_{kj})=0$. We further define the coherence lengths
as Giunti:1991sx ; Beuthe:2002ej ; Kayser:2010pr ; deGouvea:2020hfl

$L^{\rm coh}_{jk}=\frac{4\sqrt{2}E^{2}}{|\Delta m_{jk}^{2}|}\sigma\,,$ (2)

which depend on the neutrino energy and the mass-squared differences. We
assume all decoherence effects to be encoded in a single parameter $\sigma$,
which can be interpreted as the width of the neutrino wave-packet and has
dimensions of length. Estimates for the typical value of $\sigma$ depend on
the physics responsible for neutrino production and vary by orders of
magnitude. This physics contains several distance-scales, many of which
potentially inform $\sigma$. For instance, distance scales associated with
electron antineutrinos detected via inverse beta-decay may include: i) the
typical size of the beta-decaying nuclei ($\sim 10^{-5}$ nm) ii) typical
interatomic spacing that characterizes the fuel ($\sim 0.1-1$ nm for Uranium)
iii) the inverse of the antineutrino energy ($\sim 10^{-4}$ nm). Without
adding to the discussion of estimating $\sigma$, we choose to stay agnostic
and consider $\sigma$ as a free parameter, generically assuming that it will
depend on the features of neutrino production and detection.

In the presence of decoherence effects, the density matrix $\rho_{jk}$
describing the flavor content of the reactor antineutrinos produced in nuclear
power plants as a function of $L$ and $E$ is

$\tilde{\rho}_{jk}(L,E)=\tilde{U}_{ej}^{*}\tilde{U}_{ek}\exp[-i\tilde{\Delta}_{jk}]\exp[-\tilde{\xi}_{jk}(L,E)]\,,$
(3)

with

$\tilde{\Delta}_{jk}\equiv 2\pi\frac{L}{\tilde{L}^{\rm
osc}_{jk}}\equiv\frac{\Delta\tilde{m}^{2}_{jk}L}{2E}\,,$ (4)

where we have included matter effects (assuming the antineutrinos propagate
through a medium with constant density) by substituting all the quantities in
vacuum with the corresponding well-known effective matter-quantities
Giunti:1991sx . The tilde in the variables defined above denotes that a
quantity is affected by matter effects. In particular, $\tilde{\xi}_{jk}$ is
as defined in Eq. (1) but with the mass-squared difference replaced by its
matter counterpart. The electron antineutrino survival probability in presence
of decoherence effects (for a constant matter profile) is given by the $ee$
diagonal element of the density matrix:

$\displaystyle
P^{\text{dec}}(\overline{\nu}_{e}\to\overline{\nu}_{e})=\sum_{j,k}|\tilde{U}_{ej}|^{2}|\tilde{U}_{ek}|^{2}\,\exp[-i\tilde{\Delta}_{jk}-\tilde{\xi}_{jk}]\,.$
(5)

As expected, we recover the standard oscillation probability when the damping
factors $\tilde{\xi}_{jk}\rightarrow 0$ or, equivalently, when $\tilde{L}^{\rm
coh}_{jk}\rightarrow\infty$ ($\sigma\rightarrow\infty$).

We compute the oscillation parameters in matter with the help of the
parameterization discussed in Ref. Denton:2016wmg , which has been shown to be
among the most precise and most efficient ones Parke:2019vbs . The matter
effects, in principle, render the current reactor neutrino experiments
sensitive to the neutrino mass ordering. In practice, however, the sensitivity
is completely negligible. Current analyses of the world neutrino data slightly
prefer the normal mass ordering deSalas:2020pgw ; Capozzi:2017ipn ;
Esteban:2020cvm ; Kelly:2020fkv  so, for concreteness, we assume it to be
normal (i.e., $\Delta m^{2}_{31}>0$). Had we performed the analysis assuming
the inverted ordering, we would have obtained the same results. A little more
quantitatively, on the Earth’s crust, the matter potential
$\sqrt{2}G_{F}n_{e}\sim 10^{-7}$ eV2/MeV, where $n_{e}$ is the electron number
density. This is to be compared with $\Delta m^{2}_{ij}/2E$. For reactor
antineutrinos, $\Delta m^{2}_{21}/2E\gtrsim 5\times 10^{-6}$ eV2/MeV, twenty
times larger ($|\Delta m^{2}_{31}|$, of course, is thirty times larger than
$\Delta m^{2}_{21}$). Hence, matter effects will impact oscillations at, very
roughly, the 5% level in experiments, like KamLAND, sensitive to $\Delta
m^{2}_{21}$. This is especially true of KamLAND, where very long baselines
($L={\cal O}$(1000 km)) are relevant, keeping in mind that
$\sqrt{2}G_{F}n_{e}\times 1000~{}\rm km\sim 0.3$.

In order to illustrate the impact of decoherence on reactor antineutrino
oscillations at the different reactor experiments, we depict in Fig. 1 (left)
the expected electron antineutrinos survival probability for $L=1$ km,
representative of the baselines of the Daya Bay and JUNO experiment, while in
Fig. 1 (right) we depicted the average electron antineutrino survival
probability at KamLAND. In the latter, we perform a weighted average taking
into account the different baselines and reactor power outputs. We fix all
standard oscillation parameters to the current best-fit values extracted in
Ref. deSalas:2020pgw . The green, solid curve corresponds to the standard
neutrino oscillation scenario without decoherence, while the red and black
dashed ones are obtained assuming non-trivial decoherence effects associated
to $\sigma=2\times 10^{-4}$ nm and $\sigma=1\times 10^{-4}$ nm, respectively.
These values are consistent with the lower limits obtained in the current work
(see Sec. IV) and in our previous analysis of RENO + Daya Bay data
deGouvea:2020hfl , respectively. The impact of a finite value for $\sigma$ is
clear in Fig. 1. Decoherence “erases” the oscillatory behavior of the survival
probability and its impact is more pronounced at relatively smaller energies.

Figure 1: Left: the electron antineutrino oscillation probability as a
function of the neutrino energy for $L=1$ km, representative of short-baseline
experiments like RENO and Daya Bay. Right: the average electron antineutrino
oscillation probability at KamLAND. In both panels, the colors correspond to
different values of the decoherence parameter $\sigma$. The light blue,
vertical line sets the threshold for inverse beta-decay detection. All other
oscillation parameters are fixed at their current best-fit values, see
deSalas:2020pgw , and we assume that the mass ordering is normal.

##  III Data analysis

We analyze data from three different reactor experiments. RENO and Daya Bay
are reactor neutrino experiments in South Korea and China, respectively, that
measure the flux of antineutrinos from nuclear reactors at baselines $L\sim
100$ m and $L\sim 1$ km, using information from both the near and far
detectors to measure the electron antineutrino survival probability, Eq. (5).
We rely on the experimental data presented in Refs. Adey:2018zwh  and
jonghee_yoo_2020_4123573 . The procedure followed in this paper builds upon
our previous analysis†††See also Ref. An:2016pvi  for an analysis with a
smaller Daya Bay dataset. described in Ref. deGouvea:2020hfl , with the only
difference that here we also allow for the values of $\Delta m_{21}^{2}$ and
$\theta_{12}$ to vary in the fit. We refer the reader to our previous work
deGouvea:2020hfl  for more technical details and for the definition of the
$\chi^{2}$ functions $\chi^{2}_{\text{RENO}}$ and $\chi^{2}_{\text{DB}}$.

KamLAND was a long-baseline reactor experiment, placed at the site of the
former Kamiokande experiment, measuring antineutrinos from more than 50
reactor cores and distances ranging from $\sim 100$ km to $\sim 1000$ km. As
discussed earlier, because of the long baselines, matter effects are not
negligible at KamLAND, and are included in our calculations, as detailed in
Sec. II. We follow the analysis and include the data presented in Refs.
Gando:2010aa ; kamland_web . The $\chi^{2}$ function for KamLAND is

$\chi^{2}_{\text{KL}}(\vec{p})=\min_{\vec{\alpha}}\left\\{\sum_{i=1}^{N_{\text{KL}}}\left(\frac{N_{\text{dat},i}-N_{\text{exp},i}(\vec{p},\vec{\alpha})}{\sigma^{\text{KL}}_{i}}\right)^{2}+\frac{(N^{\text{tot}}_{\text{dat}}-N^{\text{tot}}_{\text{exp}}(\vec{p},\vec{\alpha}))^{2}}{N^{\text{tot}}_{\text{dat}}}+\sum_{k}\left(\frac{\alpha_{k}-\mu_{k}}{\sigma_{k}}\right)^{2}\right\\}\,,$
(6)

where the index $i$ runs over the energy bins. $N_{\text{dat},i}$ are the
observed event numbers, while $N_{\text{exp},i}(\vec{p},\vec{\alpha})$ are the
expected event numbers for a given set of oscillation parameters $\vec{p}$.
Following the collaboration approach, we include a penalty term on the total
number of events. The last term contains penalty factors for all of the
systematic uncertainties $\alpha_{k}$ with expectation value $\mu_{k}$ and
standard deviation $\sigma_{k}$. We include several sources of systematic
uncertainties, accounting for reactor uncertainties (normalizations related to
the different reactors and an uncorrelated shape error) and detector
uncertainties (detection efficiency and energy scale). We use GLoBES
Huber:2004ka ; Huber:2007ji  to compute the event numbers and to perform the
statistical analysis. The reactor fluxes are parameterized as in Ref.
Huber:2004xh  and the inverse beta-decay cross section is taken from Ref.
Vogel:1999zy . After analyzing the data from each experiment independently, we
also combine the three data sets using

$\chi^{2}_{\text{COMB}}(\vec{p})=\chi^{2}_{\text{RENO}}(\vec{p})+\chi^{2}_{\text{DB}}(\vec{p})+\chi^{2}_{\text{KL}}(\vec{p})\,.$
(7)

In order to validate our treatment of the data sets, we first assume a
perfectly coherent source and compare our results to those published by the
experimental collaborations Adey:2018zwh ; jonghee_yoo_2020_4123573 ;
Gando:2010aa . Hence, we first consider the case $\vec{p}=(\Delta
m_{31}^{2},\theta_{13},\Delta m_{21}^{2},\theta_{12})$. Next, to explore the
impact of decoherence effects, we allow for the possibility of a finite wave-
packet width $\sigma$ and consider an extended set of parameters,
$\vec{p}=(\Delta m_{31}^{2},\theta_{13},\Delta
m_{21}^{2},\theta_{12},\sigma)$.

##  IV Results

We start by comparing the $\Delta\chi^{2}$ profiles obtained from the
combination of all experiments in the presence of decoherence effects with
those obtained assuming a perfectly coherent source. The reduced $\chi^{2}$
profiles for all four accessible oscillation parameters are depicted in Fig.
2. In all plots, we marginalize over all absent parameters, including $\sigma$
when decoherence effects are allowed in the fit. The best-fit values for the
standard neutrino oscillation parameters are not strongly affected by the
possible loss of coherence. In particular, the best fit values of the mass
squared differences $\Delta m^{2}_{21}$ and $\Delta m^{2}_{31}$ are identical
regardless of whether decoherence effects are allowed in the fit. Remember
that we assume $\Delta m^{2}_{31}$ to be positive. The reduced-$\chi^{2}$
functions are somewhat shallower when decoherence effects are allowed in the
fit, as expected. The best-fit values for $\sin^{2}\theta_{12}$ and
$\sin^{2}\theta_{13}$ are instead slightly altered, moving towards slightly
larger values of $\sin^{2}2\theta_{12,13}$. Also in this case, and as
expected, the reduced-$\chi^{2}$ functions are somewhat shallower when
decoherence effects are allowed in the fit. Fig. 2 (top,left) is a reminder
that reactor antineutrino experiments are mostly sensitive to
$\sin^{2}2\theta_{12}$ and cannot really distinguish $\theta_{12}$ from
$\pi/2-\theta_{12}$. This degeneracy is resolved by solar neutrino data
PhysRevD.54.2048 ; deGouvea:1999xe ; deGouvea:2000pqg , which also provides a
more precise determination of $\sin^{2}\theta_{21}$ relative to reactor
experiments‡‡‡There is a similar degeneracy when it comes to the determination
of $\sin^{2}\theta_{13}$; values close to one, however, are very safely ruled
out by the remainder of the world’s neutrino oscillation data.. Note that the
small asymmetry in $\sin^{2}\theta_{12}$ is due to the inclusion of matter
effects in our analysis. In summary, the determination of the standard
oscillation parameters is not substantially impacted by the possible loss of
coherence of neutrino oscillations due to neutrino wave-packet separation.

Figure 2:  The reduced $\chi^{2}$ profiles as a function of the standard
neutrino oscillation parameters, obtained from the combined analysis RENO +
Daya Bay + KamLAND including decoherence effects (green, solid) and assuming a
perfectly coherent source (black, dashed). The profiles are calculated with
respect to the global minimum for each case.

Fig. 3 depicts the $\sin^{2}\theta_{12}$-$\Delta m_{21}^{2}$ (left) and
$\sin^{2}\theta_{13}$-$\Delta m_{31}^{2}$ (right) regions of parameter space
consistent with the combined data sets (filled regions in orange at 90% CL,
blue at 95% CL, green at 99% CL). In all plots, we marginalize over all absent
parameters, including $\sigma$ when decoherence effects are allowed in the
fit. The figure also depicts the allowed contours corresponding to the
analysis performed assuming a perfectly coherent source (black empty curves,
dot-dashed at 90% CL, dashed at 95% CL, solid at 99% CL). The best-fit points
from the standard analyses are indicated with black dots while the best-fit
value from the analyses including nontrivial $\sigma$ are marked with red
stars. Decoherence effects shift the allowed regions towards larger values of
$\sin^{2}2\theta_{12,13}$ and lead to larger (i.e., larger area) allowed
regions at the same CL. In the left panel, nonetheless, note that the allowed
region for $\sin^{2}\theta_{12}$ becomes a bit smaller when including
decoherence effects (see also Fig. 2 (top,left)).

Figure 3:  90, 95 and 99% CL (2 d.o.f.) allowed regions in the
$\sin^{2}\theta_{12}$-$\Delta m_{21}^{2}$ (left) and
$\sin^{2}\theta_{13}$-$\Delta m_{31}^{2}$ (right) planes from our combined
analysis of RENO + Daya Bay + KamLAND data including decoherence (filled
regions, red stars) and assuming a perfectly coherent source (black empty
contours, black dots).

In deGouvea:2020hfl , relying only on RENO + Daya Bay data, we found a
relatively stronger correlation among the parameters, with larger values of
$\sin^{2}\theta_{13}$ or smaller values of $\Delta m_{31}^{2}$ compensating
for, respectively, the decoherence effects of flattening the oscillation
maximum due to the presence of $\sigma$ or shifting the position of the first
oscillation maximum to smaller $L/E$ values. Here, the combination of data
from short-baseline experiments together with those from KamLAND significantly
reduces the allowed region in the $\sin^{2}\theta_{13}$-$\Delta m_{31}^{2}$
(compare Fig. 3 (right) with Fig. 2 of Ref. deGouvea:2020hfl ). This is a
consequence of the fact that KamLAND is more sensitive to nontrivial $\sigma$
effects that RENO and Daya Bay, as we turn to momentarily. Data from KamLAND,
it turns out, excludes the relatively small values of $\sigma$ that allow for
the stronger correlations we reported in deGouvea:2020hfl .

When it comes to the so-called solar parameters, we do not observe significant
correlations between $\sigma$ and $\Delta m_{21}^{2}$. This translates into a
lack of unexpected correlations in Fig. 3 (left). The main reason for this is
that KamLAND observes both an oscillation maximum and a minimum, reducing the
ability of compensating for decoherence effects by changing the value of
$\Delta m^{2}_{21}$. Values of $\sigma$ that are small enough to “flatten” the
minimum (which flattens faster than the maximum, as illustrated in Fig. 1) are
excluded, and we end up with a robust measurement of $\Delta m_{21}^{2}$. This
is unlike the situation at Daya Bay and RENO, which instead observe only the
first oscillation minimum.

Fig. 4 depicts the reduced $\chi^{2}$ as a function of $\sigma$, relative to
the minimum value. Marginalizing over the standard oscillation parameters, we
obtain the following best-fit value for the reactor-antineutrino-wave-packet
width: $\sigma=3.35\times 10^{-4}$ nm. The no-decoherence hypothesis,
$\sigma\to\infty$, however, is safely allowed at 90% CL and we can infer a
lower bound on $\sigma$: $\sigma>2.08\times 10^{-4}$ nm at 90% CL. This is
stronger by a factor 2 relative to the previous lower bound $\sigma>1.02\times
10^{-4}$ nm deGouvea:2020hfl , obtained by combining data only from RENO and
Daya Bay. As mentioned earlier, this explains why we do not observe strong
correlations in the $\sigma$-$\sin^{2}\theta_{13}$-$\Delta m_{31}^{2}$
parameter “volume.” While KamLAND data are virtually blind to
$\sin^{2}\theta_{13}$ and $\Delta m_{31}^{2}$, they translated into a
relatively stronger bound on $\sigma$ which in turn breaks the degeneracy
observed in Ref. deGouvea:2020hfl . KamLAND data also exclude, at the 90% CL,
the value of $\sigma$ preferred by the data from Daya Bay and RENO.

Figure 4:  The reduced $\chi^{2}$ as a function of $\sigma$ relative to its
minimum value, obtained from the combined analysis of RENO, Daya Bay and
KamLAND (red, solid) and from the combined analysis of only short-baseline
experiments (grey, dashed).

##  V Conclusions

We computed the effects of wave-packet-separation decoherence on the current
data on the oscillations of reactor antineutrinos, obtained for $L\sim$ 1 km
(Daya Bay and RENO) and for $L\sim$ 100+ km (KamLAND). We found that the
current data can exclude wave-packet sizes $\sigma<2.08\times 10^{-4}$ nm at
90% CL, assuming that neutrinos from all nuclear-reactor cores can be
characterized by the same $\sigma$. We also studied the impact of allowing for
arbitrary values of $\sigma$ when measuring $\sin^{2}\theta_{12}$,
$\sin^{2}\theta_{13}$, $\Delta m^{2}_{21}$, and $|\Delta m^{2}_{31}|$. We
found that, given the existing reactor data, these measurements are robust,
i.e., regardless of whether nontrivial $\sigma$ values are allowed in the fit,
the extracted best-fit values and error bars are approximately the same and no
unusual correlations are induced by allowing for nontrivial $\sigma$ values.

We found that KamLAND data are more sensitive to decoherence effects than
those of Daya Bay and RENO combined. This is not a trivial statement. Daya Bay
and RENO have accumulated more statistics. Furthermore, KamLAND “sees”
neutrinos from a plurality of nuclear cores and averaging-out effects, which
tend to mimic those of decoherence, are very significant. It turns out,
however, that the access to very long baselines, the relatively large value of
$\sin^{2}2\theta_{12}$, and the fact that KamLAND “sees” both oscillation
maxima and minima leads to stronger sensitivity. In the next few years, we
expect an order-of-magnitude better sensitivity from the JUNO experiment
deGouvea:2020hfl ; Cheng:2020jje .

As in our previous publication deGouvea:2020hfl , here we also choose not to
add to the very interesting but subtle discussion of expectations for $\sigma$
given antineutrinos produced in nuclear-reactor cores. We reiterate that naive
estimates are safely larger than the experimental bound obtained here.
Nonetheless, we find it is important to test the hypothesis that nuclear
reactors are, for modern practical applications, a coherent source of
antineutrinos, to probe how large decoherence effects could be, and to
understand how these might impact our ability to measure fundamental physics
parameters with reactor neutrino oscillation experiments.

## Acknowledgments

We thank the participants and organisers of “The Magnificent CE$\nu$NS
workshop 2020” for interesting conversations and comments that inspired this
work. The work of AdG is supported in part by the DOE Office of Science award
#DE-SC0010143. VDR acknowledges financial support by the SEJI/2020/016 grant
(project “Les Fosques”) funded by Generalitat Valenciana, by the Universitat
de València through the sub-programme “ATRACCIÓ DE TALENT 2019” and partial
support by the Spanish grant FPA2017-85216-P. CAT is supported by the research
grant “The Dark Universe: A Synergic Multimessenger Approach” number
2017X7X85K under the program “PRIN 2017” funded by the Ministero
dell’Istruzione, Università e della Ricerca (MIUR).

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2420222129557

# Series acceleration formulas obtained from
experimentally discovered hypergeometric
recursions

Paul Levrie1,2  John Campbell3  UAntwerpen, Belgium
KU Leuven, Belgium
York University, Canada

(2022-05-14; 2022-11-08; 2022-12-04)

###### Abstract

In 2010, Kh. Hessami Pilehrood and T. Hessami Pilehrood introduced generating
function identities used to obtain series accelerations for values of
Dirichlet’s $\beta$ function, via the Markov–Wilf–Zeilberger method. Inspired
by these past results, together with related results introduced by Chu et al.,
we introduce a variety of hypergeometric recurrences. We prove these
recurrences using the WZ method, and we apply these recurrences to obtain
series acceleration identities. We introduce a family of summations
generalizing a Ramanujan-type series for $\frac{1}{\pi^{2}}$ due to Guillera,
and a family of summations generalizing an accelerated series for Catalan’s
constant due to Lupaş, and many related results.

###### keywords:

computer proof, Wilf–Zeilberger method, series acceleration, hypergeometric
series

##  1 Introduction

In a recent article by Campbell (2022b), the hypergeometric transform

${}_{3}F_{2}\\!\\!\left[\begin{matrix}a,b,1\vspace{1mm}\\\ c,2\end{matrix}\
\Bigg{|}\
{1}\right]=-\frac{c-1}{(a-1)(b-1)}+\frac{c-1}{(a-1)(b-1)}{}_{2}F_{1}\\!\\!\left[\begin{matrix}a-1,b-1\vspace{1mm}\\\
c-1\end{matrix}\ \Bigg{|}\ {1}\right]$ (1)

was proved using the Wilf–Zeilberger (WZ) method Petkovšek et al. (1996) and
was applied to formulate a new proof of Euler’s formula $\zeta(2)=\pi^{2}/6$.
The identity in (1) was also used in relation to some problems on binomial-
harmonic sums given by Wang and Chu (2022). In this article, we provide a
number of hypergeometric identities resembling (1) that we obtained
experimentally. We prove them with the WZ method Petkovšek et al. (1996), and
use them to formulate and apply series acceleration methods. Many previously
published closed forms for fast converging series by many different authors,
including Ramanujan (1914), Guillera (2003, 2008), Hessami Pilehrood and
Hessami Pilehrood (2008, 2010), and Lupaş (2000), are special cases of our
telescoping-based series acceleration techniques.

Our methods are related to the work of Catalan (1867) and Fabry (2020) on
series evaluations, and to the first author’s previous work concerning series
by Wallis, Forsyth, and Ramanujan Levrie and Nimbran (2018); also, see
Ojanguren (2018). Catalan and Fabry used a method due to Kummer (1837) to
accelerate the convergence of some series. The repeated application of
Kummer’s transformation leads to new series. One of the examples given by
Fabry is based on the series

$\frac{\pi^{2}}{12}=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{2}}.$ (2)

This example was later taken over by Knopp (1990) in his classic text, where
we find the following exercise (Knopp, 1990, 129.c), p. 272): Prove that

$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{2}(n+1)^{2}\ldots(n+p-1)^{2}}=\frac{5p+2}{4(p+1)p!^{2}}-\frac{p(p+1)^{3}}{4}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{2}(n+1)^{2}\ldots(n+p+1)^{2}}.$

Using this recurrence with $p=1$ transforms the series in (2) into a new
series that converges faster. Repeated applications of the recurrence lead to
the series

$\frac{\pi^{2}}{12}=\frac{7}{8}-\frac{8}{4}\cdot\left(\frac{17}{576}-\frac{192}{4}\cdot\left(\frac{27}{345600}-\frac{1080}{4}\cdot\left(\ldots\right)\right)\right)=\frac{1}{4}\sum_{n=1}^{\infty}(-1)^{n-1}\frac{10n-3}{n^{2}(2n-1)\binom{2n}{n}}$

with convergence rate $\tfrac{1}{4}$ (Fabry, 2020, p.138). The above
recurrence can be rewritten and generalized in terms of hypergeometric
functions. This result is given in the second part of Theorem 5 below.

While our hypergeometric recurrences given in Sections 2–5 are of interest in
their own right, the main results of this article are our new telescoping
proofs of known identities equating fundamental constants with fast convergent
series, together with related evaluations for new, fast converging series. We
provide, for example, new proofs and generalizations of remarkable formulas
for $\frac{1}{\pi^{2}}$ that were discovered and proved by Guillera Guillera
(2003, 2008) and that are indicated in Table 1. It is remarkable how the WZ
proof certificates for many of our recurrences are hugely complicated compared
to the elegance and simplicity of our closed forms obtained from these
recurrences. However, the WZ proofs are straightforward compared to
Fabry/Kummer-like proofs of these results. These require some ingenuity and
are all based on special identities enabling a telescoping effect, as we
explain in more detail in Section 6.

###  1.1 Related literature

Much of this article is inspired by the methods and results introduced by
Hessami Pilehrood and Hessami Pilehrood (2010), who introduced a number of
fast convergent series for Catalan’s constant
$G=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}$. These series are indicated
in Table 1, which also includes a number of other previously published series
involved in or otherwise related to our work.

Constant | Summand | Convergence rate
---|---|---
$G$ | $\frac{1}{8}\frac{2^{4n}(6n-1)}{n^{2}(2n-1)\binom{4n}{2n}\binom{2n}{n}}$ | $\frac{1}{4}$
$G$ | $-\frac{1}{64}\frac{(-2^{8})^{n}(40n^{2}-24n+3)}{n^{3}(2n-1)\binom{4n}{2n}^{2}\binom{2n}{n}}$ | $\frac{1}{4}$
$G$ | $-\frac{1}{2}\frac{(-2^{3})^{n}(3n-1)}{n^{3}\binom{2n}{n}^{3}}$ | $\frac{1}{8}$
$G$ | $\frac{1}{64}\frac{2^{8n}(580n^{2}-184n+15)}{n^{3}(2n-1)\binom{6n}{3n}\binom{6n}{4n}\binom{4n}{2n}}$ | $\frac{4}{729}$
$\pi^{2}$ | $2\frac{2^{4n}(3n-1)}{n^{3}\binom{2n}{n}^{3}}$ | $\frac{1}{4}$
$\pi^{2}$ | $4\frac{2^{4n}(n+1)(3n+1)}{n(2n+1)^{2}\binom{2n}{n}^{3}}$ | $\frac{1}{4}$
$\pi^{2}$ | $6\frac{21n-8}{n^{3}\binom{2n}{n}^{3}}$ | $\frac{1}{64}$
$\pi^{-1}$ | $\frac{1}{2}(-2^{-6})^{n}\binom{2n}{n}^{3}(4n+1)$ | $1$
$\pi^{-1}$ | $\frac{1}{16}\left(2^{-12}\right)^{n}\binom{2n}{n}^{3}(42n+5)$ | $\frac{1}{64}$
$\pi^{-2}$ | $\frac{1}{8}\left(-2^{-12}\right)^{n}\binom{2n}{n}^{5}\left(20n^{2}+8n+1\right)$ | $\frac{1}{4}$
$\pi^{-2}$ | $\frac{1}{128}\left(-{2^{-20}}\right)^{n}\binom{2n}{n}^{5}\left(820n^{2}+180n+13\right)$ | $\frac{1}{1024}$
Table 1: Previously known series involved in or otherwise related to our work.
For each of the series indicated in this table, we let it be understood that
the lower limit is $n=1$ if the denominator of the corresponding summand
vanishes for $n=0$ and that the lower limit is $n=0$ otherwise. References for
the above series are given, respectively, as follows: Hessami Pilehrood and
Hessami Pilehrood (2010), Campbell (2022a); Hessami Pilehrood and Hessami
Pilehrood (2010); Lima (2012); Lupaş (2000), Guillera (2008); Lima (2012),
Hessami Pilehrood and Hessami Pilehrood (2010), Guillera (2008); Hessami
Pilehrood and Hessami Pilehrood (2010), Campbell (2022a), Hessami Pilehrood
and Hessami Pilehrood (2008), Bauer (1859), Ramanujan (1914), Guillera (2003),
and Guillera (2008).

The second formula for $G$ shown in Table 1 is used in the Wolfram Language to
calculate $G$ Marichev et al. . This motivates the techniques and applications
given in this article, in which a new acceleration method is applied to
generalize this formula for $G$ and many other fast convergent series
formulas.

Our series recursions and accelerations are also related to the work by Chu
(2006) and Chu and Zhang (2014). In Chu and Zhang (2014), the series
rearrangement referred to as the modified Abel lemma on summation by parts was
employed to obtain hypergeometric series recursions related to our work. For a
hypergeometric series $h(x,y)$ involving the variables $x$ and $y$ in its
parameters, we prove recursions such as

$h(x,y)=r_{1}(x,y)+r_{2}(x,y)\,h(x,y+1)$ (3)

for rational functions $r_{1}(x,y)$ and $r_{2}(x,y)$, and we then apply such
recursions iteratively, yielding accelerated series formulas. In contrast to
Chu and Zhang (2014), we do not make use of any Abel-type summation results.
Instead, we employ the WZ method. Furthermore, our experimental approaches
allow us to obtain recursions for ${}_{p}F_{q}(-1)$-families. It seems that
${}_{p}F_{q}(-1)$-series are not considered in past references such as Chu and
Zhang (2014). Also, an advantage of our WZ-based methodologies is due to how
the WZ difference equation

$F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k)$

may be manipulated in order to obtain new and nontrivial hypergeometric
identities, using the WZ pairs corresponding to our WZ proofs, together with
WZ-based series manipulations as in Campbell (2021). It appears that the Maple
implementation of the WZ method cannot be applied to the recursions due to Chu
and Zhang in Chu and Zhang (2014). This shows how the methods in Chu and Zhang
(2014) are nontrivially different from ours, which rely on the WZ method.

###  1.2 Organization of the article

The rest of our article is mainly structured and organized in the manner
indicated as follows. In each of the following main sections, apart from
Section 6 and the Conclusion, we introduce recursions for families of
${}_{p}F_{q}(x)$-series for fixed $p$, $q$, and $x$. The recurrences are given
in terms of the parameters of the generalized hypergeometric functions. We
present WZ proofs of these recursions, and then apply these recursions to
obtain series acceleration identities, to obtain fast convergent series
formulas for fundamental constants. Given one such fast convergent series, our
method naturally gives rise to an infinite family, by altering the input
parameters. In this regard, we choose to emphasize, as in Sections 3.1 and 5.1
below, our infinite families of generalizations of especially remarkable
results due to Lupaş (2000) and Guillera (2003).

In Section 2, we introduce experimentally discovered
${}_{3}F_{2}(1)$-recurrences, yielding formulas for very fast convergent
series, such as

$\zeta(2)=\frac{7}{4}-2\sum_{n=2}^{\infty}\frac{(3n-1)(7n-5)}{n^{2}(n-1)^{2}\binom{2n}{n}^{3}},$
(4)

a series with convergence rate $\frac{1}{64}$.

In Section 3, we introduce experimentally discovered
${}_{3}F_{2}(-1)$-recurrences, yielding an infinite family of series
generalizing the algorithm used to compute $G$ in the Wolfram Language
Marichev et al. .

In Section 4, we introduce an experimentally discovered
${}_{5}F_{4}(-1)$-recurrence, yielding new WZ proofs of two of Ramanujan’s
series for $\frac{1}{\pi}$, along with the new formula

$G=\frac{1}{16}\sum_{n=1}^{\infty}\frac{2^{6n}(336n^{4}-464n^{3}+216n^{2}-42n+3)}{n^{3}(2n-1)^{3}\binom{4n}{2n}^{3}},$
(5)

which has a convergence rate of $\frac{1}{64}$.

In Section 5, we introduce experimentally discovered
${}_{6}F_{5}(1)$-recurrences, which we apply to obtain a variety of results,
including new proofs/generalizations of Guillera’s series for
$\frac{1}{\pi^{2}}$.

In Section 6, we briefly describe the experimental mathematics-based nature
about how we had discovered the hypergeometric recurrences proved via the WZ
method in this article.

In Section 7, we conclude by briefly describing some further areas of research
based on the techniques given in this article.

##  2 Experimentally discovered ${}_{3}F_{2}(1)$-recurrences

For the sake of brevity, we assume familiarity with WZ theory Petkovšek et al.
(1996), generalized hypergeometric series, the Pochhammer symbol, etc. Our
first main result is highlighted as Theorem 1 below.

A number of the hypergeometric recurrences introduced in this article are of
the form indicated in (3). As below, we are letting the summand of a
hypergeometric series $h(x,y)$ as in (3) be denoted as
$\text{summand}(x,y,k)$. This notation will be useful in our generalizing a
fast converging series formula for $\frac{1}{\pi^{2}}$ due to Guillera, as in
Section 5.1.

######  Lemma 1.

If the recurrence in (3) holds, then the infinite series

$\sum_{k=0}^{\infty}\text{\emph{summand}}(x,y,k)$ (6)

may be expressed as

$\sum_{j=0}^{m-1}\left(\prod_{i=0}^{j-1}r_{2}(x,y+i)\right)r_{1}(x,y+j)+\left(\prod_{i=0}^{m-1}r_{2}(x,y+i)\right)\sum_{k=0}^{\infty}\text{\emph{summand}}(x,y+m,k)$

for natural numbers $m$, under the assumption that the above series converge,
for given values of $x$ and $y$.

###### Proof.

This follows in a direct way from repeated applications of (3). ∎

Ideally, we would want that

$\left(\prod_{i=0}^{m-1}r_{2}(x,y+i)\right)\sum_{k=0}^{\infty}\text{summand}(x,y+m,k)\to
0$

as $m\to\infty$. Although the above vanishing condition does not necessarily
imply acceleration, we would, ideally, want, from Lemma 1, to obtain a series

$\sum_{j=0}^{\infty}\left(\prod_{i=0}^{j-1}r_{2}(x,y+i)\right)r_{1}(x,y+j)$

that is new, convergent, and faster convergent relative to (6).

By symmetry, given a recursion such as
$h^{\prime}(x,y)=r_{1}^{\prime}(x,y)+r_{2}^{\prime}(x,y)h^{\prime}(x+1,y)$ for
a hypergeometric series $h^{\prime}$, we may again apply Lemma 1, by setting
$h(x,y)$ as $h^{\prime}(y,x)$. We may similarly generalize Lemma 1 using
recurrences of the form
$h^{\prime\prime}(x,y)=r_{1}^{\prime\prime}(x,y)+r_{2}^{\prime\prime}(x,y)\,h^{\prime\prime}(x,y+\mu)$
for fixed $\mu\in\mathbb{N}$. For the time being, we proceed by providing and
proving a new ${}_{3}F_{2}(1)$-recursion of the form indicated in (3).

######  Theorem 1.

The ${}_{3}F_{2}(1)$-transform

$\displaystyle{}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x+1,1\vspace{1mm}\\\
x+y,x+y+1\end{matrix}\ \Bigg{|}\ {1}\right]=\frac{2x+3y-1}{2(2y-1)}$
$\displaystyle\hskip
142.26378pt+\frac{(y-1)y(y+1)}{2(2y-1)(x+y)(x+y+1)}\cdot{}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x+1,1\vspace{1mm}\\\
x+y+1,x+y+2\end{matrix}\ \Bigg{|}\ {1}\right]$

holds true for $x$ and $y$ such that both sides of the above equality
converge.

###### Proof.

We set $x=-n$. So, it remains to prove that

$\displaystyle\sum_{k=0}^{\infty}\frac{(1-n)_{k}(-n)_{k}}{(y-n)_{k}(-n+y+1)_{k}}\cdot$
$\displaystyle\frac{k^{2}(2-4y)+k\left(n(8y-4)-8y^{2}+2\right)+n^{2}(2-4y)+n\left(8y^{2}-2\right)-3y^{3}-2y^{2}+y}{(2n-3y+1)(k-n+y)(k-n+y+1)}$

constantly equals $1$. Now, we restrict $n$ to being a member of the set
$\mathbb{N}_{0}$. In this case, Pochhammer symbols of the form $(-n)_{k}$
vanish for $k>n$, with $k\in\mathbb{N}_{0}$. So, for $n\in\mathbb{N}_{0}$, we
may replace the upper limit of the above series with $n$. Now, we may apply
the WZ method, giving us a WZ proof certificate that is given explicitly in
the first draft available via the https://hal.archives-
ouvertes.fr/hal-03659300/ site. So, using the WZ method, we have shown that
the above infinite series identity holds for $n\in\mathbb{N}_{0}$. We may thus
apply Carlson’s theorem (Bailey, 1935, p. 39) as in Ekhad and Zeilberger
(1994) to show that the same series identity holds for real $n$. ∎

######  Theorem 2.

The ${}_{3}F_{2}(1)$-transform

${}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x+1,1\vspace{1mm}\\\
x+y,x+y+1\end{matrix}\ \Bigg{|}\
{1}\right]=1+\frac{x(x+1)}{(x+y)(x+y+1)}\cdot{}_{3}F_{2}\\!\\!\left[\begin{matrix}x+1,x+2,1\vspace{1mm}\\\
x+y+1,x+y+2\end{matrix}\ \Bigg{|}\ {1}\right]$

holds true for $x$ and $y$ such that both sides of the above equality
converge.

###### Proof.

Again, we set $x=-n$. So, it remains to prove that

$\sum_{k=0}^{\infty}\frac{(1-n)_{k}(-n)_{k}}{(y-n)_{k}(-n+y+1)_{k}}\frac{y(2k-2n+y+1)}{(k-n+y)(k-n+y+1)}=1.$

Again, we consider the case for $n\in\mathbb{N}_{0}$, so that we may again
replace the upper parameter with $n$. Again, we may apply the WZ method. The
corresponding WZ proof certificate is given below:

$\frac{k(k-n+y)(k-n+y+1)\left(-2kn+ky-k+2n^{2}-2ny+2n-y+1\right)}{(k-n-1)(k-n)(n-y)(n-y+1)(2k-2n+y+1)}.$

We may thus apply Carlson’s theorem, as in our proof of Theorem 1. ∎

###  2.1 A series acceleration method

Let us define

$s(x,y)=\frac{x}{x+y}\cdot{}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x+1,1\vspace{1mm}\\\
x+y,x+y+1\end{matrix}\ \Bigg{|}\
{1}\right]=\sum_{n=0}^{\infty}\frac{(x)_{n}(x)_{n+1}}{(x+y)_{n}(x+y)_{n+1}}\
.$

We rewrite the recurrences given as Theorems 1 and 2, respectively, so that

$s(x,y)=\frac{x(2x+3y-1)}{2(2y-1)(x+y)}+\frac{(y-1)y(y+1)}{2(2y-1)(x+y)^{2}}s(x,y+1)$
(7)

and that

$s(x,y)=\frac{x}{x+y}+\frac{x^{2}}{(x+y)^{2}}s(x+1,y).$ (8)

Note that we know the value of $s(x,y)$ in some cases:

$s(1,2)=7-\frac{2}{3}\pi^{2},s(\tfrac{1}{2},2)=3-\frac{9}{32}\pi^{2},s(1,\tfrac{3}{2})=10-3\pi,s(\tfrac{1}{2},\tfrac{3}{2})=2-\frac{16}{3\pi}.$
(9)

Indeed, we have, for instance, that:

$s(1,2)=\sum_{n=0}^{\infty}\frac{4}{(n+3)(n+2)^{2}(n+1)}=\sum_{n=0}^{\infty}\left(\frac{2}{n+1}-\frac{2}{n+3}-\frac{4}{(n+2)^{2}}\right),$

leading to the given closed form for $s(1,2)$. We can deal with the second one
in a similar manner. For the third and fourth ones, we can use a closed form
for the partial sum of the series, as in:

$s(\tfrac{1}{2},\tfrac{3}{2})=\sum_{n=1}^{\infty}\frac{2}{(n+1)(2n-1)}\frac{\binom{2n}{n}^{2}}{2^{4n}}\
\ \ \mbox{with}\ \ \
\sum_{n=1}^{k-1}\frac{2}{(n+1)(2n-1)}\frac{\binom{2n}{n}^{2}}{2^{4n}}=2-\frac{8}{3}\frac{k(4k-1)}{2k-1}\frac{\binom{2k}{k}^{2}}{2^{4k}}.$

We then take the limit and use Wallis’s product formula for $\pi$.

We can use the $s$-recurrences above to accelerate the convergence of the
corresponding series. Iterating the first recurrence leads to the following
accelerated series, via Lemma 1:

$s(x,y)=\sum_{n=0}^{\infty}\frac{x(2x+3(y+n)-1)}{2(2(y+n)-1)(x+y+n)}\frac{(y-1)_{n}(y)_{n}(y+1)_{n}}{2^{2n}(y-\frac{1}{2})_{n}(x+y)^{2}_{n}}.$

For the four special values given above, we find the following four series:

$\displaystyle\zeta(2)$
$\displaystyle=\frac{7}{4}-\sum_{n=2}^{\infty}\frac{3n+1}{(n+1)n^{2}(n-1)\binom{2n}{n}},$
$\displaystyle\zeta(2)$
$\displaystyle=\frac{16}{9}-\sum_{n=2}^{\infty}\frac{2^{4n}}{(2n+1)n(n-1)\binom{2n}{n}^{3}},$
(10) $\displaystyle\pi$
$\displaystyle=-2\sum_{n=0}^{\infty}\left(2^{-4}\right)^{n}\binom{2n}{n}\frac{6n+5}{(2n+3)(2n+1)(2n-1)},$
$\displaystyle\frac{1}{\pi}$
$\displaystyle=-12\sum_{n=0}^{\infty}\left(2^{-8}\right)^{n}\binom{2n}{n}^{3}\frac{n^{2}}{(2n-1)(2n-3)}.$

The fast convergent series evaluation in (10) is closely related to the series
for $\pi^{2}$ from Campbell (2022a) shown in Table 1, noting the convergence
rate of $\frac{1}{4}$ in both cases.

If we use (8) in (7), we get a recurrence relating $s(x,y)$ to $s(x+1,y+1)$,
and other accelerated sums, including (4). This series is especially
remarkable for many reasons, perhaps most notably due to its very fast
convergence rate of $\frac{1}{64}$. In the same manner that we have obtained
(4), the following formulas may be proved:

$\displaystyle\zeta(2)$
$\displaystyle=\frac{16}{9}-2\sum_{n=1}^{\infty}\frac{2^{4n}(14n^{2}+11n+1)(n+1)}{n(4n+3)(4n+1)^{2}(2n-1)\binom{2n}{n}\binom{4n}{2n}^{2}},$
$\displaystyle\pi$
$\displaystyle=\frac{10}{3}-2\sum_{n=1}^{\infty}\frac{(14n+11)(2n+1)(3n+1)\binom{2n}{n}}{(4n+3)(4n+1)^{2}(2n-1)\binom{4n}{2n}^{2}},$
$\displaystyle\frac{1}{\pi}$
$\displaystyle=-\frac{1}{8}\sum_{n=0}^{\infty}\left(2^{-12}\right)^{n}\binom{2n}{n}^{3}\frac{(6n+1)(14n-3)(2n+1)}{(2n-1)^{2}}.$

This leads to an infinite number of possibilities.

##  3 Experimentally discovered ${}_{3}F_{2}(-1)$-recurrences

In view of the closed forms shown in (9), it is not clear as to how to obtain
accelerated series for Catalan’s constant
$G=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{2}}$. This motivates the
exploration of alternating variants and generalizations Theorems 1 and 2.
Defining $t(x,y)$ so that

$t(x,y)=\sum_{n=0}^{\infty}\frac{(x)_{n}(x)_{n+1}}{(x+y)_{n}(x+y)_{n+1}}(-1)^{n},$
(11)

we find that $t\big{(}\frac{1}{2},2\big{)}=\frac{3}{8}(6G-5)$. Note that

$t(x,y)=\frac{x}{x+y}\cdot{}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x+1,1\vspace{1mm}\\\
x+y,x+y+1\end{matrix}\ \Bigg{|}\ {-1}\right].$

This motivates the use of analogues of our above acceleration methods, in
order to obtain accelerated series for $G$.

The recurrence highlighted below is remarkable in the sense that it is far
from clear as to how our proofs for the preceding Theorems could be mimicked
or reformulated so as to prove the identity highlighted as Theorem 3.

######  Theorem 3.

The ${}_{3}F_{2}(-1)$-recurrence

$t(x,y)=\frac{x(2x^{2}+6xy+5y^{2}+2x+4y)}{4(x+y)^{2}(x+y+1)}-\frac{y^{2}(y+1)(y+2)}{4(x+y)^{2}(x+y+1)^{2}}t(x,y+2)$
(12)

holds true for suitably bounded $x$ and $y$.

###### Proof.

Rewrite $x$ as $-n$. So, it remains to show that

$\sum_{k=0}^{\infty}\frac{(-1)^{k}(-n)_{k}(-n)_{k+1}}{(-n+y)_{k}(-n+y)_{1+k}}\left(-\frac{(n-y)^{2}(-n+y+1)\left(\frac{(y+1)(y+2)y^{2}}{(k-n+y)(k-n+y+1)^{2}(k-n+y+2)}+4\right)}{n\left(-6ny+2(n-1)n+5y^{2}+4y\right)}\right)$

constantly equals $1$. In line with our previous WZ proofs, we temporarily
restrict our attention to the case whereby $n\in\mathbb{N}_{0}$. Again, the
vanishing of $(-n)_{k}$ for $k>n\in\mathbb{N}_{0}$ produces a finite sum. A WZ
proof certificate for this finite sum is given in the first article draft
available via the HAL site previously indicated. Once again, we may apply
Carlson’s theorem. ∎

It is common to denote quotients of Pochhammer symbols so that

$\\!\\!\left[\begin{matrix}\alpha_{1},\ldots,\alpha_{p}\vspace{1mm}\\\
\beta_{1},\ldots,\beta_{q}\end{matrix}\right]_{n}=\frac{\left(\alpha_{1}\right)_{n}\cdots\left(\alpha_{p}\right)_{n}}{\left(\beta_{1}\right)_{n}\cdots\left(\beta_{q}\right)_{n}},$

as in the series acceleration identity shown below.

######  Theorem 4.

The expression $t(x,y)$ equals the accelerated infinite series

$\displaystyle\sum_{j=0}^{\infty}$
$\displaystyle\left(-2^{-2}\right)^{j}\frac{xy^{2}(y+1)(y+2)\left(6x(2j+y)+5(2j+y)^{2}+4(2j+y)+2x^{2}+2x\right)}{4(x+y)^{2}(x+y+1)^{2}(2j+x+y)^{2}(2j+x+y+1)}\cdot$
$\displaystyle\\!\\!\left[\begin{matrix}\frac{y}{2}+1,\frac{y}{2}+1,\frac{y}{2}+2,\frac{y+3}{2}\vspace{1mm}\\\
\frac{x+y}{2}+1,\frac{x+y}{2}+1,\frac{x+y+3}{2},\frac{x+y+3}{2}\end{matrix}\right]_{j-1}$

if both sides converge for $x$ and $y$.

###### Proof.

Through repeated applications of the recurrence given as Theorem 3, we may
rewrite $t(x,y)$ as

$\displaystyle\sum_{j=0}^{m-1}\frac{(-1)^{j}\left(\prod_{i=0}^{j-1}\frac{(2i+y)^{2}(1+2i+y)(2+2i+y)}{4(2i+x+y)^{2}(1+2i+x+y)^{2}}\right)x\left(2x+2x^{2}+4(2j+y)+6x(2j+y)+5(2j+y)^{2}\right)}{4(2j+x+y)^{2}(1+2j+x+y)}$
$\displaystyle+(-1)^{m}\left(\prod_{i=0}^{m-1}\frac{(2i+y)^{2}(1+2i+y)(2+2i+y)}{4(2i+x+y)^{2}(2i+x+y+1)^{2}}\right)t(x,y+2m).$

Taking $m\to\infty$, it is easily seen that the latter term vanishes. ∎

######  Example 1.

Setting $x=\frac{1}{2}$ and $y=1$ in Theorem 4, we obtain the accelerated
series identity

$8-4\pi=\sum_{j=1}^{\infty}\frac{\left(-2^{4}\right)^{j}\left(40j^{2}-12j-1\right)\binom{2j}{j}}{j(2j-1)^{2}(4j+1)\binom{4j}{2j}^{2}}.$

The above series is of convergence rate $\frac{1}{4}$.

###  3.1 A family of accelerated series for $G$

By setting $x=-\frac{1}{2}$ and $y=2$ in Theorem 4, or by setting
$x=\frac{1}{2}$ and $y=2$ in Theorem 4, in either case, this gives us an
equivalent version of the series used to calculate $G$ in the Wolfram Language
Marichev et al. . For example, in the latter case, we obtain that

$\frac{40}{3}-16G=\sum_{j=1}^{\infty}\frac{\left(-2^{8}\right)^{j}\left(40j^{2}+28j+3\right)}{j^{2}(4j+1)^{2}(4j+3)\binom{2j}{j}\binom{4j}{2j}^{2}},$
(13)

and adding 16 times the series given by the second formula for $G$ in Table 1
gives us a series with partial sums that are evaluable in closed form.

######  Example 2.

Set $x=\frac{3}{2}$ and $y=2$ in Theorem 4. This gives us

$\frac{16G}{3}-\frac{664}{135}=\sum_{j=1}^{\infty}\frac{\left(-2^{8}\right)^{j}\left(40j^{2}+52j+15\right)}{j^{2}(4j+1)^{2}(4j+3)^{2}(4j+5)\binom{2j}{j}\binom{4j}{2j}^{2}}.$

######  Example 3.

Set $x=\frac{5}{2}$ and $y=2$ in Theorem 4. This gives us that

$\frac{23048}{70875}-\frac{16G}{45}=\sum_{j=1}^{\infty}\frac{\left(-2^{8}\right)^{j}\left(40j^{2}+76j+35\right)}{j^{2}(4j+1)^{2}(4j+3)^{2}(4j+5)^{2}(4j+7)\binom{2j}{j}\binom{4j}{2j}^{2}}.$

Continuing in this manner gives us an infinite family of extensions of (13).
It appears that our techniques for generating such families of accelerated
series have not been considered previously, as in past literature related to
Hessami Pilehrood and Hessami Pilehrood (2010), including the work by Adell
and Lekuona (2016, 2015); Hessami Pilehrood and Hessami Pilehrood (2011);
Hessami Pilehrood et al. (2012, 2014); Wang and Xu (2021).

###  3.2 Further ${}_{3}F_{2}(-1)$-recurrences

######  Theorem 5.

The ${}_{3}F_{2}(-1)$-recurrences

$\displaystyle{}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x,1\vspace{1mm}\\\
x+y,x+y\end{matrix}\ \Bigg{|}\
{-1}\right]=1-\frac{x^{2}}{(x+y)^{2}}\cdot{}_{3}F_{2}\\!\\!\left[\begin{matrix}x+1,x+1,1\vspace{1mm}\\\
x+1+y,x+1+y\end{matrix}\ \Bigg{|}\ {-1}\right]$
$\displaystyle{}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x,1\vspace{1mm}\\\
x+y,x+y\end{matrix}\ \Bigg{|}\
{-1}\right]=\frac{2x^{2}+6xy+5y^{2}+y}{4(x+y)^{2}}$ $\displaystyle\hskip
142.26378pt-\frac{y(y+1)^{3}}{4(x+y)^{2}(x+y+1)^{2}}\cdot{}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x,1\vspace{1mm}\\\
x+y+2,x+y+2\end{matrix}\ \Bigg{|}\ {-1}\right]$

hold true, if all of the above series converge.

###### Proof.

Write $x=-n$. We may mimic our previous WZ proofs, via Carlson’s theorem, to
prove the first out of the two ${}_{3}F_{2}(-1)$-recurrences shown above. A
corresponding WZ certificate is shown below:

$\frac{ky(k-n+y)^{2}\left(-2kn+ky-2k+2n^{2}-2ny+4n-2y+2\right)}{(k-n-1)^{2}(n-y+1)^{2}\left(2k^{2}-4kn+2ky+2n^{2}-2ny+y^{2}\right)}.$

We may apply the same approach to prove the latter ${}_{3}F_{2}(-1)$-formula
under consideration. A WZ proof certificate is given explicitly in the
previously provided HAL URL. ∎

With the notation

$s(x,y)={}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x,1\vspace{1mm}\\\
x+y,x+y\end{matrix}\ \Bigg{|}\ {-1}\right],$

we may express the recurrences in Theorem 5 as follows:

$\displaystyle s(x,y)$ $\displaystyle=1-\frac{x^{2}}{(x+y)^{2}}s(x+1,y),$
$\displaystyle s(x,y)$
$\displaystyle=\frac{2x^{2}+6xy+5y^{2}+y}{4(x+y)^{2}}-\frac{y(y+1)^{3}}{4(x+y)^{2}(x+y+1)^{2}}s(x,y+2).$

If we use the second recurrence to calculate
$s(\tfrac{1}{2},2)=\frac{9}{2}\,(1-\frac{\pi}{4})$, $s(1,2)=4\,(3-4\ln(2))$,
and $s(\tfrac{1}{2},1)=G$, we find the corresponding series in Lupaş (Lupaş,
2000, p. 6). If we combine the recurrences to write $s(x,y)$ as a function of
$s(x+1,y+2)$ and of $s(x+2,y+2)$, we obtain, for $x=\frac{1}{2}$ and $y=1$, a
series for Catalan’s constant proved in Hessami Pilehrood and Hessami
Pilehrood (2010).

##  4 Experimentally discovered ${}_{5}F_{4}(-1)$-recurrences

######  Theorem 6.

The recurrences

$\displaystyle\left(x+\frac{y-1}{2}\right)\cdot{}_{5}F_{4}\\!\\!\left[\begin{matrix}x,x,x,x+\frac{y+1}{2},1\vspace{1mm}\\\
x+y,x+y,x+y,x+\frac{y-1}{2}\end{matrix}\ \Bigg{|}\
{-1}\right]=x+\frac{y-1}{2}$
$\displaystyle-\frac{x^{3}}{(x+y)^{3}}\left(x+1+\frac{y-1}{2}\right)\cdot{}_{5}F_{4}\\!\\!\left[\begin{matrix}x+1,x+1,x+1,x+1+\frac{y+1}{2},1\vspace{1mm}\\\
x+1+y,x+1+y,x+1+y,x+1+\frac{y-1}{2}\end{matrix}\ \Bigg{|}\ {-1}\right]$
$\displaystyle\left(x+\frac{y-1}{2}\right)\cdot{}_{5}F_{4}\\!\\!\left[\begin{matrix}x,x,x,x+\frac{y+1}{2},1\vspace{1mm}\\\
x+y,x+y,x+y,x+\frac{y-1}{2}\end{matrix}\ \Bigg{|}\
{-1}\right]=\frac{x-1}{2}+y$
$\displaystyle-\frac{y^{3}}{(x+y)^{3}}\left(x+\frac{y}{2}\right)\cdot{}_{5}F_{4}\\!\\!\left[\begin{matrix}x,x,x,x+\frac{y+2}{2},1\vspace{1mm}\\\
x+y+1,x+y+1,x+y+1,x+\frac{y}{2}\end{matrix}\ \Bigg{|}\ {-1}\right]$

hold true, if all of the above series converge.

###### Proof.

Again, we set $x=-n$, and we may apply the WZ method and Carlson’s theorem in
much the same way as before. WZ proof certificates for each out of the two
recursions shown above are given explicitly in via the previously provided HAL
site. ∎

With the notation:

$s(x,y)=\left(x+\frac{y-1}{2}\right)\cdot{}_{5}F_{4}\\!\\!\left[\begin{matrix}x,x,x,x+\frac{y+1}{2},1\vspace{1mm}\\\
x+y,x+y,x+y,x+\frac{y-1}{2}\end{matrix}\ \Bigg{|}\ {-1}\right],$

we find that:

$\displaystyle s(x,y)$
$\displaystyle=x+\frac{y-1}{2}-\frac{x^{3}}{(x+y)^{3}}s(x+1,y),$
$\displaystyle s(x,y)$
$\displaystyle=\frac{x-1}{2}+y-\frac{y^{3}}{(x+y)^{3}}s(x,y+1).$

Note that for $x=\tfrac{1}{2},y=\tfrac{1}{2}$, $s(x,y)$ is the Bauer–Ramanujan
series for $\frac{1}{\pi}$ of convergence rate $1$ shown in Table 1. If we try
to accelerate this series with one of the recurrences, we get the
Bauer–Ramanujan series again in both cases. Hence no acceleration. However, if
we use them one after the other, then we obtain Ramanujan’s series for
$\frac{1}{\pi}$ of convergence rate $\frac{1}{64}$, which is indicated in
Table 1. For $x=\tfrac{1}{2},y=1$, we have that

$s(x,y)=\frac{1}{2}\sum_{n=0}^{\infty}(-1)^{n}\frac{1}{(2n+1)^{2}}=\frac{G}{2}.$

Using both recurrences one after the other leads to (5). For $x=1,y=1$, we
obtain, in this case, the series for $\pi^{2}$ of convergence rate
$\frac{1}{64}$ indicated in Table 1.

##  5 Experimentally discovered ${}_{6}F_{5}(1)$-recurrences

######  Theorem 7.

The recurrences

$\displaystyle\left(x+\frac{y-1}{2}\right)\cdot{}_{6}F_{5}\\!\\!\left[\begin{matrix}x,x,x,x,x+\frac{y+1}{2},1\vspace{1mm}\\\
x+y,x+y,x+y,x+y,x+\frac{y-1}{2}\end{matrix}\ \Bigg{|}\
{1}\right]=x+\frac{y-1}{2}$
$\displaystyle+\frac{x^{4}}{(x+y)^{4}}\left(x+1+\frac{y-1}{2}\right)\cdot{}_{6}F_{5}\\!\\!\left[\begin{matrix}x+1,x+1,x+1,x+1,x+1+\frac{y+1}{2},1\vspace{1mm}\\\
x+1+y,x+1+y,x+1+y,x+1+y,x+1+\frac{y-1}{2}\end{matrix}\ \Bigg{|}\ {1}\right]$
$\displaystyle\left(x+\frac{y-1}{2}\right)\cdot{}_{6}F_{5}\\!\\!\left[\begin{matrix}x,x,x,x,x+\frac{y+1}{2},1\vspace{1mm}\\\
x+y,x+y,x+y,x+y,x+\frac{y-1}{2}\end{matrix}\ \Bigg{|}\
{1}\right]=\frac{5y^{2}+6(x-1)y+2(x-1)^{2}}{4(2y-1)}$
$\displaystyle-\frac{y^{5}}{2(2y-1)(x+y)^{4}}\left(x+\frac{y}{2}\right)\cdot{}_{6}F_{5}\\!\\!\left[\begin{matrix}x,x,x,x,x+\frac{y+2}{2},1\vspace{1mm}\\\
x+y+1,x+y+1,x+y+1,x+y+1,x+\frac{y}{2}\end{matrix}\ \Bigg{|}\ {1}\right]$

hold true, if all of the above series converge.

###### Proof.

Again, we write $x=-n$, and we may again mimic our WZ-based proofs given as
above. WZ proof certificates for each case are given explicitly in the HAL
site previously given. ∎

###  5.1 Generalizations of Guillera’s series

With the notation

$s(x,y)=\left(x+\frac{y-1}{2}\right)\cdot{}_{5}F_{4}\\!\\!\left[\begin{matrix}x,x,x,x,x+\frac{y+1}{2},1\vspace{1mm}\\\
x+y,x+y,x+y,x+y,x+\frac{y-1}{2}\end{matrix}\ \Bigg{|}\ {1}\right],$

Theorem 7 gives us that

$\displaystyle s(x,y)$
$\displaystyle=x+\frac{y-1}{2}+\frac{x^{4}}{(x+y)^{4}}s(x+1,y),$
$\displaystyle s(x,y)$
$\displaystyle=\frac{5y^{2}+6(x-1)y+2(x-1)^{2}}{4(2y-1)}-\frac{y^{5}}{2(2y-1)(x+y)^{4}}s(x,y+1).$

Note that for $x=\tfrac{1}{2},y=\tfrac{3}{2}$, $s(x,y)$ can be calculated from
its partial sums:

$s(\tfrac{1}{2},\tfrac{3}{2})=\frac{1}{4}\sum_{n=0}^{\infty}\left(2^{-8}\right)^{n}\binom{2n}{n}^{4}\frac{4n+3}{(n+1)^{4}}=4-\frac{32}{\pi^{2}}$

If we try to accelerate this series with the first of the recurrences, we get
the same series. With the second one, we get Guillera’s formula for
$\frac{1}{\pi^{2}}$ introduced in Guillera (2003) and reproduced in Table 1.

If we use both the recurrences for $x=1,y=1$, we obtain a series for
$\zeta(3)$ given in Lupaş’s paper Lupaş (2000). Now, let us further simplify
our expression for the latter recursion given in Theorem 7, letting $h$,
$r_{1}$, and $r_{2}$ be as below:

$\displaystyle\text{summand}(x,y,k)=\frac{\left(x+\frac{y-1}{2}\right)(x)_{k}^{4}\left(x+\frac{y+1}{2}\right)_{k}}{\left(x+\frac{y-1}{2}\right)_{k}(x+y)_{k}^{4}},$
$\displaystyle r_{1}(x,y)=\frac{6(x-1)y+2(x-1)^{2}+5y^{2}}{4(2y-1)},$
$\displaystyle r_{2}(x,y)=-\frac{y^{5}}{2(2y-1)(x+y)^{4}}.$

So, we obtain that

$\sum_{k=0}^{\infty}\text{summand}(x,y,k)=r_{1}(x,y)+r_{2}(x,y)\sum_{k=0}^{\infty}\text{summand}(x,y+1,k),$
(14)

leading us to the following. A direct application of Lemma 1, for
$\text{summand}(x,y,k)$, $r_{1}(x,y)$, and $r_{2}(x,y)$ as above, may be
regarded as a generalization of Guillera’s series for $\frac{1}{\pi^{2}}$
introduced in Guillera (2003) and reproduced in Table 1, in view of our
derivation of this formula along with the following results.

######  Example 4.

Setting $x=\frac{3}{2}$ and $y=\frac{3}{2}$, our series acceleration formula
given in Lemma 1 yields the following:

$\frac{128}{\pi^{2}}=\sum_{j=0}^{\infty}\left(-2^{-12}\right)^{j}\binom{2j}{j}^{5}\frac{20j^{2}+32j+13}{(j+1)^{4}}.$

######  Example 5.

Setting $x=\frac{5}{2}$ and $y=\frac{3}{2}$, Lemma 1 gives us that:

$\frac{2048}{81\pi^{2}}=\sum_{j=0}^{\infty}\left(-2^{-12}\right)^{j}\binom{2j}{j}^{5}\frac{20j^{2}+56j+41}{(j+1)^{4}(j+2)^{4}}.$

######  Example 6.

Setting $x=\frac{1}{2}$ and $y=1$, Lemma 1 yields

$28\zeta(3)=\sum_{j=1}^{\infty}\frac{\left(-2^{8}\right)^{j}\left(-10j^{2}+6j-1\right)}{j^{5}\binom{2j}{j}^{5}}.$

######  Example 7.

If we use both recurrences from Theorem 7 one after the other, with
$x=\frac{3}{2}$ and $y=\frac{3}{2}$, we obtain Guillera’s series for
$\frac{1}{\pi^{2}}$ from Guillera (2008) reproduced in Table 1. It has
convergence rate $\frac{1}{2^{10}}$.

##  6 On the experimental discovery of hypergeometric transforms

In their classic text, Borwein et al. (2004) describe how a human-computer
dialogue is central to experimental mathematics (Borwein et al., 2004, p.
vii). This is exemplified by the use of computers to explore new areas in
mathematics, and via the development of a better understanding of mathematical
phenomena using computer-based results and computer-based tools (Borwein et
al., 2004, p. vii–viii). In this regard, we have used a computer-based
approach to study the problem of accelerating series, and our computer
explorations have led us to identify patterns that show us that many
previously published results by many different authors, including Ramanujan
(1914), Guillera (2003, 2008), Hessami Pilehrood and Hessami Pilehrood (2008,
2010), and Lupaş (2000), are special cases of our methods. We find it
appropriate to explain how we had obtained and discovered our main results in
an experimental way.

In 2018 Levrie and Nimbran (2018), the recurrence

${}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x,1\vspace{1mm}\\\
x+y,x+y\end{matrix}\ \Bigg{|}\
{1}\right]=\frac{2x+3y-2}{2(2y-1)}+\frac{y^{3}}{2(2y-1)(x+y)^{2}}\cdot{}_{3}F_{2}\\!\\!\left[\begin{matrix}x+1,x+1,1\vspace{1mm}\\\
x+1+y,x+1+y\end{matrix}\ \Bigg{|}\ {1}\right]$

was introduced and proved. If we write

$s(x,y)={}_{3}F_{2}\\!\\!\left[\begin{matrix}x,x,1\vspace{1mm}\\\
x+y,x+y\end{matrix}\ \Bigg{|}\
{1}\right]=\sum_{n=0}^{\infty}\frac{(x)_{n}^{2}}{(x+y)_{n}^{2}}$ (15)

and then write $f(x,y,n)=\frac{2n+2x-y}{2(2y-1)}$, we find that this summand
may be manipulated in the below indicated manner, starting with the reindexing
argument shown below, and noting that a telescoping argument is implicitly
used to simplify the below expression involving $f(x,y,n)$ and $f(x,y,n+1)$:

$\displaystyle s(x,y)$
$\displaystyle=1+\sum_{n=0}^{\infty}\frac{(x)_{n+1}^{2}}{(x+y)_{n+1}^{2}}$
$\displaystyle=1+\sum_{n=0}^{\infty}\frac{(x)_{n}^{2}}{(x+y)_{n}^{2}}\cdot\frac{\left(x+n\right)^{2}}{\left(x+y+n\right)^{2}}$
$\displaystyle=1+\sum_{n=0}^{\infty}\frac{(x)_{n}^{2}}{(x+y)_{n}^{2}}\left(f(x,y,n)-f(x,y,n+1)\frac{\left(x+n\right)^{2}}{\left(x+y+n\right)^{2}}+\frac{y^{3}}{2\left(2y-1\right)\left(x+y+n\right)^{2}}\right)$
$\displaystyle=1+f(x,y,0)+\frac{y^{3}}{2\left(2y-1\right)}\sum_{n=0}^{\infty}\frac{(x)_{n}^{2}}{(x+y)_{n+1}^{2}}$
$\displaystyle=\frac{2x+3y-2}{2(2y-1)}+\frac{y^{3}}{2\left(2y-1\right)(x+y)^{2}}\sum_{n=0}^{\infty}\frac{(x)_{n}^{2}}{(x+y+1)_{n}^{2}}.$

Our approach toward obtaining and proving hypergeometric recurrences is
similar relative to the above derivation and involves computer-based
experiments with summands as in (15). Our research is based on explorations
involving the use of Computer Algebra Systems to experiment with
hypergeometric series in order to emulate the telescoping argument involving
$f(x,y,n)$, as above. This has led us toward patterns based on our
computational experiments that generalize closed forms due to Ramanujan,
Lupaş, and Guillera, and others. This is motivated by open problems concerning
Ramanujan’s and Guillera’s series, such as the problem as to how it may be
possible to generalize Guillera’s series so as to obtain hypergeometric
expansions for $\frac{1}{\pi^{k}}$ for higher powers, with of $k>2$ Guillera
(2020). As below, we offer some brief descriptions as to how we had discovered
the hypergeometric transforms given in this article.

For Theorem 1, a similar approach was used, relative to the above
manipulations of the summand for (15), and the rational function identity
given below was involved:

$\displaystyle\frac{(x+n)(x+n+1)}{(x+y+n)(x+y+n+1)}=$
$\displaystyle\frac{2n+2x-y+1}{2(2y-1)}-\frac{2n+2+2x-y+1}{2(2y-1)}\frac{(x+n)(x+n+1)}{(x+y+n)(x+y+n+1)}$
$\displaystyle+\frac{(y-1)y(y+1)}{2(2y-1)}\frac{1}{(x+y+n)(x+y+n+1)}.$

Similarly, for Theorem 5, summand manipulations were applied to

$s(x,y)=\sum_{n=0}^{\infty}(-1)^{n}\frac{(x)_{n}^{2}}{(y)_{n}^{2}}$

so as to obtain the identity

$\displaystyle(x+n)^{2}=$
$\displaystyle\text{summand}(x,y,n)+\text{summand}(x,y,n+1)\frac{(x+n)^{2}}{(x+y+n+1)^{2}}+\frac{y(1+y)^{3}}{4(x+y+n+1)^{2}}$

with $\text{summand}(x,y,n)=\frac{2n^{2}+2(2x+y)n+2x^{2}+2xy-y^{2}-y}{4}$.
Similar approaches led to our discoveries for the remaining Theorems in this
article.

##  7 Conclusion

In an article due to Guillera (2020), conjectured Ramanujan-like series for
$\frac{1}{\pi^{k}}$ for $k\geq 3$ were recorded, such as the formula

$\displaystyle\sum_{n=0}^{\infty}\frac{\left(\frac{1}{2}\right)_{n}^{5}\left(\frac{1}{3}\right)_{n}\left(\frac{2}{3}\right)_{n}\left(\frac{1}{4}\right)_{n}\left(\frac{3}{4}\right)_{n}}{\left(1\right)_{n}^{9}}(4528n^{4}+3180n^{3}+972n^{2}+147n+9)\left(-\frac{27}{256}\right)^{n}$
$\displaystyle=\frac{768}{\pi^{4}}$

conjectured by Zhao. Guillera also proved a bilateral series expansion for
negative powers of $\pi$ Guillera (2020). This was achieved via the
periodicity of the function

$f(x)=e^{-i\pi x}(\cos\pi x)^{2j+1}\prod_{s_{k}\neq\frac{1}{2}}\frac{\cos\pi
x-\cos\pi s_{k}}{1-\cos\pi s_{k}}\sum_{n=-\infty}^{\infty}A(n,x)$

for a function $A(n,x)$ of the form

$A(n,x)=\left(\prod_{i=0}^{2m}\frac{(s_{i})_{n+x}}{(1)_{n+x}}\right)\sum_{k=0}^{m}a_{k}(n+x)^{k}z_{0}^{n+x}.$

We encourage the pursuit of research based on how our recursive and WZ-based
methods may be applied to obtain series for $\frac{1}{\pi^{k}}$ for higher
values of $k>2$. In this regard, the main obstacle is due to how difficult it
is to obtain series for $\frac{1}{\pi^{k>2}}$ that may be accelerated
according to our methods. For example, there are classically known series for
$\frac{1}{\pi^{2}}$ that may be easily proved inductively and that may be
accelerated using our hypergeometric transforms, but it is not obvious as to
how it may be possible to start with a relatively “simple” series for
$\frac{1}{\pi^{3}}$ or $\frac{1}{\pi^{4}}$ and then apply our evaluation
technique. One possibility to consider might be based on manipulations of the
Cauchy product of known series for $\frac{1}{\pi^{2}}$, and we leave it to a
separate project to investigate this option.

Consider the two recurrences for the hypergeometric expression $s(x,y)$
involved in Section 5.1. As above, we have applied these recurrences one after
the other. If we instead apply these recurrences so as to express $s(x,y)$ in
terms of $s(x+3,y+3)$, then, by mimicking our proof of Guillera’s series from
Guillera (2008) shown in Table 1, we can show that

$\frac{8}{\pi^{2}}=\sum_{j=0}^{\infty}\left(-2^{-60}\right)^{j}\binom{6j}{3j}^{5}\frac{p(j)}{(2j-1)^{5}(6j-5)^{3}(6j-1)^{5}},$
(16)

where $p(j)$ denotes the following polynomial.

    -125 + 950*j + 31660*j^2 - 50024*j^3 - 14995984*j^4 +
    235669685116640*j^5 - 5284712547631680*j^6 + 52653694429319040*j^7 -
    306803615328753408*j^8 + 1157357544632934912*j^9 -
    2952738590842027008*j^10 + 5159445973725726720*j^11 -
    6097655344343592960*j^12 + 4666164796084690944*j^13 -
    2088682533545361408*j^14 + 415520397881671680*j^15

The above series for $\frac{1}{\pi^{2}}$ has a convergence rate of
$\frac{1}{1073741824}$. Formulas as in (16) may also be determined by applying
series $n$-sections to the accelerated series given in this article, as
explained below, and we encourage the investigation of this subject.

Let us begin, for example, with the series for $\pi^{2}$ from Campbell (2022a)
recorded in Table 1. By applying a series bisection to this series, we find
that

$\displaystyle\frac{\pi^{2}}{4}=\sum_{n=1}^{\infty}\frac{2^{8n}(2n+1)(6n+1)}{2n(4n+1)^{2}\binom{4n}{2n}^{3}}+\sum_{n=1}^{\infty}\frac{2^{8n}n(3n-1)}{4(4n-1)^{2}(2n-1)\binom{4n-2}{2n-1}^{3}},$

and a reindexing-type argument then gives us that

$\pi^{2}=\sum_{n=1}^{\infty}2^{8n}\frac{240n^{4}-8n^{3}-40n^{2}-n+1}{n^{2}(2n-1)(4n+1)^{2}\binom{4n}{2n}^{3}},$
(17)

and this was recently proved in a different way, via the WZ method, by
Campbell (2022c). We see that (17) has an acceleration rate of $\frac{1}{16}$,
relative to the acceleration rate of $\frac{1}{4}$ in the original formula
from Campbell (2022a). Chu and Zhang used the same kind of acceleration
approach via series bisections in Example 89 in Chu and Zhang (2014), and we
may obtain similarly accelerated series via series $n$-sections, as opposed to
series bisections.

We have numerically found that

$\frac{\pi^{4}}{8}=\sum_{n=1}^{\infty}\frac{2^{8n}(3n-1)\left(7n^{2}-5n+1\right)}{n^{7}\binom{2n}{n}^{7}}.$

How can this conjectured formula be proved using our methods, or otherwise?

###### Acknowledgements.

The authors want to thank a number of expert Reviewers who have offered many
helpful suggestions that have considerably improved the presentation of our
work.

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