B(1) = +½ set 10: the André function #24024
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Luschny's paper of functions starts with the (generalised) Bernoulli
function and its relationship to the zeta function, then defines other
interesting functions in terms of earlier ones – central Bernoulli,
Genocchi, alternating Bernoulli, Worpitzky representation and Hasse
series, Euler (tangent and secant), Bernoulli secant, extended Bernoulli
and Euler…
But this whole web of definitions has to stop somewhere, and Luschny
stops at an entire function encoding both Bernoulli and Euler numbers:
the André function A(s), which itself interpolates the André numbers
(OEIS A000111, a core sequence). He writes when he finally defines it—
> Considering the long chain of definitions on which [A(s)] is based, it
> is astonishing how easily it can be represented by a single function.
> A(s) = (-i)^(s+1) Li_{-s}(i) + i^(s+1) Li_{-s}(-i)
This can itself be manipulated into a form involving zeta functions,
giving both an expression for the related *Euler zeta numbers/function*
(a sequence appearing in Euler's *De summis serierum reciprocarum*
(1735)) and a link to Dirichlet L-functions:
A(s) = 2 s! / (2π)^(s+1) * (ζ(s+1, 1/4) - ζ(s+1, 3/4) cos πs)
Z(s) = A(s) / s! [tends to 0 in the limit of large positive s]
As such we have come full circle, and the André function forms the
capstone of the Bernoulli(1) = +½ project (cf. sympy#23926).
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Benchmark results from GitHub Actions Lower numbers are good, higher numbers are bad. A ratio less than 1 Significantly changed benchmark results (PR vs master) Significantly changed benchmark results (master vs previous release) before after ratio
[41d90958] [45dcb6ca]
<sympy-1.11.1^0>
- 963±4μs 620±1μs 0.64 solve.TimeSparseSystem.time_linear_eq_to_matrix(10)
- 2.79±0.01ms 1.16±0ms 0.42 solve.TimeSparseSystem.time_linear_eq_to_matrix(20)
- 5.62±0.02ms 1.71±0ms 0.30 solve.TimeSparseSystem.time_linear_eq_to_matrix(30)
Full benchmark results can be found as artifacts in GitHub Actions |
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Okay, I think this looks good. |
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References to other Issues or PRs
Tenth and last part of the changes formerly included in #23926, building upon all nine other PRs in the set.
Brief description of what is fixed or changed
This PR introduces
$$\mathcal A(s) = (-i)^{s+1} Li_{-s}(i) + i^{s+1} Li_{-s}(-i)$$
$$\mathcal A(s) = 2 (2\pi)^{-(s+1)} (\zeta(s+1,1/4) - \cos(\pi s)\zeta(s+1,3/4)) \Gamma(s+1)$$
$$\mathcal A(s) = 2(2^s\sin(\pi s/2)\eta(-s)+\cos(\pi s/2)\beta(-s))$$
$$\mathcal A(s)=2\Re\frac{Li_{-s}(i)}{i^{s+1}}\qquad(s\in\mathbb R)$$
andre(), Luschny's entire extension of OEIS A000111:Release Notes
andrefunction implementing the entire extension of OEIS A000111 described by Luschny.