Calculate delta_kt_prime (Perez eqn 2 and eqn 3), or return a default value
    for use with :py:func:`_dirint_bins`.

def _delta_kt_prime_dirint(kt_prime, use_delta_kt_prime, times):
    """
    Calculate delta_kt_prime (Perez eqn 2 and eqn 3), or return a default value
    for use with :py:func:`_dirint_bins`.
    """
    if use_delta_kt_prime:
        # Perez eqn 2
        kt_next = kt_prime.shift(-1)
        kt_previous = kt_prime.shift(1)
        # replace nan with values that implement Perez Eq 3 for first and last
        # positions. Use kt_previous and kt_next to handle series of length 1
        kt_next.iloc[-1] = kt_previous.iloc[-1]
        kt_previous.iloc[0] = kt_next.iloc[0]
        delta_kt_prime = 0.5 * ((kt_prime - kt_next).abs().add(
                                (kt_prime - kt_previous).abs(),
                                fill_value=0))
    else:
        # do not change unless also modifying _dirint_bins
        delta_kt_prime = pd.Series(-1, index=times)
    return delta_kt_prime