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The asymptotic expansion of digamma function is a starting point for the derivation of approximants for harmonic sums or Euler-Mascheroni constant. It is usual to derive such approximations as values of logarithmic function, which leads to the expansion of the exponentials of digamma function. In this paper the asymptotic expansion of the function exp ( p ? ( x + t ) ) is derived and analyzed in details, especially for integer values of parameter p. The behavior for integer values of p is proved and as a consequence a new identity for Bernoulli polynomials. The obtained formulas are used to improve know inequalities for Euler's constant and harmonic numbers.

The aim of this paper is to extend and refine the Ramanujan–Lodge harmonic number expansion into negative powers of a triangular number. We construct a faster asymptotic series and some new sharp inequalities for the harmonic numbers. Romanian National Authority for Scientific Research/[PN-II-ID-PCE-2011–3-0087]//Rumanía Universidad de Costa Rica/[]/UCR/Costa Rica UCR::Vicerrectoría de Investigación::Unidades de Investigación::Ciencias Básicas::Centro de investigaciones Matemáticas y Metamatemáticas (CIMM)

Recently, Srivastava et al. (2011) [2] unified and extended several interesting generalizations of the familiar Hurwitz–Lerch Zeta function Φ ( z , s , a ) by introducing a Fox–Wright type generalized hypergeometric function in the kernel. For this newly introduced special function, two integral representations of different kinds are investigated here by means of a known result involving a Fox–Wright generalized hypergeometric function kernel, which was given earlier by Srivastava et al. (2011) [2] , and by applying some Mathieu ( a , λ ) -series techniques. Finally, by appealing to each of these two integral representations, two sets of two-sided bounding inequalities are proved, thereby extending and generalizing the recent work by Jankov et al. (2011) [15] .

In this paper we prove new complete monotonicity properties of some functions involving the gamma function. As consequences of them we establish various new upper and lower bounds for the gamma function and the harmonic numbers. (C) 2011 Elsevier Ltd. All rights reserved.

Let the numbers P(r,n,k) be defined by P(r,n,k):=P"r(H"n^(^1^)-H"k^(^1^),...,H"n^(^r^)-H"k^(^r^)), where P"r(x"1,...,x"r)=(-1)^rY"r(-0!x"1,-1!x"2,...,-(r-1)!x"r) and Y"r are the exponential complete Bell polynomials. By observing that the numbers P(r,n,k) generate two Riordan arrays, we establish several general summation formulas, from which series of harmonic number identities are obtained. In particular, some of these harmonic number identities also involve other special combinatorial sequences, such as the Stirling numbers of both kinds, the Lah numbers, the Bernoulli numbers and polynomials and the Cauchy numbers of both kinds.

Let γ = 0.577215 … be the Euler–Mascheroni constant, and let R n = ∑ k = 1 n 1 k − log ( n + 1 2 ) . We prove that for all integers n ≥ 1 , 1 24 ( n + a ) 2 ≤ R n − γ 1 24 ( n + b ) 2 with the best possible constants a = 1 24 [ − γ + 1 − log ( 3 / 2 ) ] − 1 = 0.55106 … and b = 1 2 . This refines the result of D. W. DeTemple, who proved that the double inequality holds with a = 1 and b = 0 .

In a recent paper Dattoli and Srivastava [3], by resorting to umbral calculus, conjectured several generating functions involving harmonic numbers. In this sequel to their work our aim is to rigorously demonstrate the truth of the Dattoli-Srivastava conjectures by making use of simple analytical arguments. In addition, one of these conjectures is stated and proved in more general form. (C) 2009 Elsevier Inc. All rights reserved.

In this paper, we consider a kind of sums involving Cauchy numbers, which have not been studied in the literature. By means of the method of coefficients, we give some properties of the sums. We further derive some recurrence relations and establish a series of identities involving the sums, Stirling numbers, generalized Bernoulli numbers, generalized Euler numbers, Lah numbers, and harmonic numbers. In particular, we generalize some relations between two kinds of Cauchy numbers and some identities for Cauchy numbers and Stirling numbers.

We use methods of umbral calculus and algebraic nature to make some progress in the theory of generating functions involving harmonic numbers. We derive old and new results on generating functions in a fairly simple way and show that the method can be applied to find a common bridge with the theory of well-known (rather classical) generating functions.

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain very simple connections between the Stirling numbers of both kinds and other generalized harmonic numbers. Further, we suggest that Riordan arrays associated with such generalized harmonic numbers allow us to find new generating functions of many combinatorial sums and many generalized harmonic number identities.

Let H n be the nth harmonic number. For all integers n ⩾ 1 we have a - log ( e 1 / ( n + 1 ) - 1 ) ⩽ H n b - log ( e 1 / ( n + 1 ) - 1 ) , with the best possible constants a = 1 + log ( e - 1 ) = 0.5672 … and b = γ = 0.5772 … . This refines a recently published result of Batir, who proved that the double-inequality holds with a = log ( π 2 / 6 ) ( = 0.4977 … ) and b = γ .

We consider summations over digamma and polygamma functions, often with summands of the form (\pm 1)^n\psi(n+p/q)/n^r and (\pm 1)^n\psi^{(m)} (n+p/q)/n^r, where m, p, q, and r are positive integers. We develop novel general integral representations and present explicit examples. Special cases of the sums reduce to known linear Euler sums. The sums of interest find application in quantum field theory, including evaluation of Feynman amplitudes., Comment: to appear in J. Comput. Appl. Math.; corrected proof available online with this journal; no figures

For a natural number n, the authors propose and develop three new series representations for the Riemann Zeta function ζ(2n+1). The infinite series occurring in each of these three representations for ζ(2n+1) converges remarkably faster than that in Wilton's result. Furthermore, one of the three series representations for ζ(2n+1) involves the most rapidly convergent series among all the hitherto known members of the family of series representations considered here. Relevant connections of the results presented in this paper with many other known series representations for ζ(2n+1) are also briefly indicated.