B(1) = +½ set 4: generalised Genocchi function #23989
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Luschny defines the generalised Genocchi function in terms of the generalised Bernoulli function (implemented in sympy#23984) as G(s, a) = 2^s (B(s, a/2) - B(s, (a+1) / 2)) which can be rather easily rewritten as G(s, a) = 2 (B(s, a) - 2^s B(s, (a+1) / 2)) which is the form used here. When s is a nonnegative integer this function reproduces the (so-called) Genocchi polynomials, which have integer coefficients; when in addition a = 1 the Genocchi numbers are obtained, with G(1) = -1. The second term in the rewritten expression (2^s B(s, (a+1)/2)) is what Luschny calls the central Bernoulli function, which reduces to the central Bernoulli numbers when a = 0.
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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] [930eb707]
- 992±1μs 625±2μs 0.63 solve.TimeSparseSystem.time_linear_eq_to_matrix(10)
- 2.89±0ms 1.17±0ms 0.41 solve.TimeSparseSystem.time_linear_eq_to_matrix(20)
- 5.76±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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As a note, the checklist at the top of #23926 includes not only my ideas for PRs, but also the ideas which have been suggested as important go-alongs, such as the moving of polynomial functionality to the polynomials module just above. |
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Looks good. |
Luschny defines the generalised Euler function in terms of the generalised Genocchi function (implemented in sympy#23989) as ⎧ ⎛a⎞ ⎛a + 1⎞ ⎪ -ψ⎜─⎟ + ψ⎜─────⎟ s = -1 ⎪ ⎝2⎠ ⎝ 2 ⎠ E(s, a) = ⎨ ⎪ -G(s + 1, a) ⎪ ───────────── s ≠ -1 ⎩ s + 1 where the top expression is the limiting value of the bottom. When s is a nonnegative integer the function reproduces the Euler polynomials, but unlike the Bernoulli and Genocchi polynomials, a must be set to ½ *and* the result must be multiplied by 2^s to obtain the Euler numbers. For even more clarity Luschny disambiguates the latter with the designation "secant", calling 2^s E(s, 1) the Euler tangent numbers.
References to other Issues or PRs
Part 4 of the changes formerly included in #23926, building upon #23984.
Brief description of what is fixed or changed
This PR generalises
$$G(s,a)=2\left(B(s,a)-2^sB\left(s,\frac{a+1}2\right)\right)$$
genocchi()to the generalised Genocchi function described in Luschny:Release Notes
genocchi()now accepts two complex arguments, implementing the generalised Genocchi function described by Luschny.