key: cord-1028518-l08pep4b
authors: Qi, Feng; Huang, Chuan-Jun
title: Computing sums in terms of beta, polygamma, and Gauss hypergeometric functions
date: 2020-08-24
journal: Rev R Acad Cienc Exactas Fis Nat A Mat
DOI: 10.1007/s13398-020-00927-y
sha: 30e3307d100a7954c970537d7c295bd689551e31
doc_id: 1028518
cord_uid: l08pep4b

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.

In [14, Theorem 1] and [23, p. 80, Eq. (7.5) ], it was obtained that The logarithmic derivative [ln (z)] = (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] [26] [27] [28] [29] and closely related references therein.

In this paper, we will extend the identities (1.1), (1.3), and (1.4), compute generalized sums in terms of the beta function B(z, w) and its partial derivatives, polygamma functions ψ (k) (z) for k = 0, 1, the Gauss hypergeometric function 2 F 1 (−n, z; 1 + z; −1), a determinant, and an identity.

In this section, via replacing corresponding integers k by a complex variable (z), we generalize identities (1.1), (1.3), and (1.4) in terms of the beta function and its partial derivatives, polygamma functions, and integrals as follows. 

It is obvious that f (0) = 0 and

Integrating on both sides of the above equality with respect to t over 

In particular, we have n q=0 n q

Proof These identities follow from differentiating with respect to z on both sides of (2.1) and (2.2) and employing partial derivatives

and

The proof of Theorem 2.3 is complete. 

It is apparent that h(0) = 0 and

Integrating the above equation with respect to t over [0, 1] arrives at

Consequently, it follows that n q=0 n q

Further differentiating m times with respect to z on both sides of (2. 

The Gauss hypergeometric function 2 F 1 (α, β; γ ; z) are defined by

for |z| < 1, for complex numbers α, β ∈ C, for γ ∈ C\{0, −1, −2, . . . }, and for

which is called the rising factorial of c ∈ C.

In the above sections, we mainly consider summability of n q=0 n q 

for (μ), (z) > 0 and arg u a < π. Taking u = μ = a = 1 and λ = n in (3.2) leads to

Hence, it follows that 2 F 1 (−n, z; 1 + z; −1) = z where

is the falling factorial of c ∈ C.

Proof By virtue of the formulas

and [1, p. 561, Item 15.4.6] or [5, p. 442, Item 18.5.7] , we obtain

In [2, p. 40, Exercise 5], it is stated that the n-th derivative of the ratio u(t) v(t) can be computed by

where u(t) and v(t) = 0 are differentiable functions, W (n+1)×(n+1) (t) denotes the determinant of the (n + 1) × (n + 1) matrix

the elements of the (n + 1) × 1 matrix U (n+1)×1 (t) are u k,1 (t) = u (k−1) (t) for 1 ≤ k ≤ n + 1, and the elements of the (n + 1)

for 1 ≤ i ≤ n +1 and 1 ≤ j ≤ n. This conclusion has been employed in the papers [6] [7] [8] [9] [10] [11] [12] [13] [15] [16] [17] [18] 20, 21, 24] and closely related references. Applying the formula (3.5) to u(w) = (w−1) n+z and v(w) = w + 1 results in

n + z 0 2 n+z 4 0 0 · · · 0 0 0 n + z 1 2 n−1+z 1 4 0 · · · 0 0 0 n + z 2 Proof Let n ≥ 2 and

where

0, otherwise.

Theorem 2.2 in [19] reads that the determinant |P n | of the tridiagonal matrix P n for n ≥ 2 can be computed explicitly by Setting α k = n+z k−1 2 k−1 for 1 ≤ k ≤ n + 1, β k = k − 1 for 2 ≤ k ≤ n + 1, and γ k = 4 for 1 ≤ k ≤ n in (3.8) arrives at

Substituting this into (3.3) leads to (3.7) . The proof of Theorem 3.3 is complete.

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Derivatives of tangent function and tangent numbers

Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials

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Some determinantal expressions and recurrence relations of the Bernoulli polynomials. Mathematics

A closed formula for the Horadam polynomials in terms of a tridiagonal determinant

Explicit formulas and identities for the Bell polynomials and a sequence of polynomials applied to differential equations

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Some properties of the Catalan-Qi function related to the Catalan numbers

Some identities for a sequence of unnamed polynomials connected with the Bell polynomials

Several formulas for special values of the Bell polynomials of the second kind and applications

A determinantal expression for the Fibonacci polynomials in terms of a tridiagonal determinant

Several explicit and recurrent formulas for determinants of tridiagonal matrices via generalized continued fractions

Some properties of the Bernoulli numbers of the second kind and their generating function

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A positive answer to Bhatia-Li conjecture on the monotonicity for a new mean in its parameter

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The authors appreciate anonymous referees for their careful corrections to and valuable comments on the original version of this paper.