Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions
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Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions Feng Qi1,2,3
· Chuan-Jun Huang4
Received: 23 February 2020 / Accepted: 17 August 2020 © The Royal Academy of Sciences, Madrid 2020
Abstract In the paper, by virtue of the binomial inversion formula, a general formula of higher order derivatives for a ratio of two differentiable function, and other techniques, the authors compute several sums in terms of the beta function and its partial derivatives, polygamma functions, the Gauss hypergeometric function, and a determinant. These results generalize known ones in combinatorics. Keywords Sum · Identity · Beta function · Polygamma function · Gauss hypergeometric function · Determinant · Binomial inversion formula · Derivative formula for a ratio of two differential functions Mathematics Subject Classification Primary 33C05; Secondary 05A10 · 11A25 · 11B65 · 33B15
Contents 1 Background and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Computing sums in terms of beta and polygamma functions . . . . . . . . . . . . . 3 Computing a sum in terms of the Gauss hypergeometric function and a determinant References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Feng Qi [email protected]; [email protected]; [email protected] https://qifeng618.wordpress.com Chuan-Jun Huang [email protected]; [email protected]
1
Institute of Mathematics, Henan Polytechnic University, Jiaozuo 454010, Henan, China
2
College of Mathematics and Physics, Inner Mongolia University for Nationalities, Tongliao 028043, Inner Mongolia, China
3
School of Mathematical Sciences, Tianjin Polytechnic University, Tianjin 300387, China
4
Department of Mathematics, Ganzhou Teachers College, Ganzhou 341000, Jiangxi, China 0123456789().: V,-vol
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F. Qi, C.-J. Huang
1 Background and motivations In [14, Theorem 1] and [23, p. 80, Eq. (7.5)], it was obtained that n n (−1)q 1 = k+n q q +k k k q=0
(1.1)
for k ≥ 1 and n ≥ 0. The binomial inversion formula in [4, p. 192, (5.48)] reads that n n n n g(n) = (−1) f () ⇐⇒ f (n) = (−1) g(). (1.2) =0
=0
Applying (1.2) into (1.1) yields n n (−1)q k = k+n q q+k
(1.3)
k
q=0
for k, n ≥ 0. This can be rewritten as n n 1 [(−1)q B(k, q + 1)] = q k+n
(1.4)
q=0
for k ≥ 1 and n ≥ 0, where
B(z, w) =
1
t z−1 (1 − t)w−1 dt, (z), (w) > 0
0
denotes the classical Euler beta function. The beta function B(z, w) and the classical Euler gamma function ∞
(z) =
t z−1 e−t dt, (z) > 0
0
have the relation B(z, w) =
(z)(w) , (x), (y) > 0. (z + w)
(z) The logarithmic derivative [ln (z)] = (z) is denoted by ψ(z) and the derivatives ψ (k) (z) for k ≥ 0 are called polygamma functions. For very recent results on the beta, gamma, and polygamma functions, please refer to the papers [22,25–29] and closely r
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