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The zeros of the digamma function are known to be simple and real, but up to now few identities involving them have appeared in the literature. By establishing a Weierstrass infinite product for a particular regularization of the digamma function, we are able to find interesting formulas for the sums of the nth powers of the reciprocals of its zeros, for n≥2. We make a parallel study of the zeros of the logarithmic derivative of the Barnes G-function. We also compare asymptotic estimates of the zeros of the digamma function and those of its Barnes G-function analogue.

We extend some known inequalities for the digamma function to the q-digamma function. Furthermore, we estimate the unique positive real root of the q-digamma function @j"q.

In this paper we shall locate a class of fundamental identities for the gamma function and trigonometric functions in the chart of functional equations for the zeta- functions as a manifestation of the underlying modular relation. We use the beta-transform but not the inverse Heaviside integral. Instead we appeal to the reciprocal relation for the Euler digamma function which gives rise to the partial fraction expansion for the cotangent function. Through this we may incorporate basic results from the theory of the digamma (and gamma) function, thereby dispensing also with the beta-transform. Section 4 could serve as a foundation of the theory of the gamma function through the digamma function.

We present some properties associated with the monotonicity and the complete monotonicity of the psi (or digamma) function and establish the higher-order estimate for the familiar Euler–Mascheroni constant γ. Relevant connections of the results presented here with those derived in earlier works are also pointed out.

The aim of this work is to find ''good'' approximations to the Digamma function @J. We construct an infinite family of ''basic'' functions {I"a,[email protected]?[0,1]} covering the Digamma function. These functions are shown to approximate @J locally and asymptotically, and it is shown that for any [email protected]?R^+, there exists an a such that @J(x)=I"a(x). Local and global bounding error functions are found and, as a consequence, new inequalities for the Digamma function are introduced. The approximations are compared to another, well-known, approximation of the Digamma function and we show that an infinite number of members of the family are better.

The definite integral M ( a ) := 4 π ∫ 0 π / 2 x 2 d x x 2 + ln 2 ⁡ ( 2 e − a cos ⁡ x ) \begin{equation*} M(a):= \frac {4}{\pi } \int _{0}^{\pi /2} \frac {x^{2} \, dx } {x^{2} + \ln ^{2}( 2 e^{-a} \cos x ) }\end{equation*} is related to the Laplace transform of the digamma function L ( a ) := ∫ 0 ∞ e − a s ψ ( s + 1 ) d s , \begin{equation*} L(a) := \int _{0}^{\infty } e^{-a s} \psi (s+1) \, ds, \end{equation*} by M ( a ) = L ( a ) + γ / a M(a) = L(a) + \gamma /a when a > ln ⁡ 2 a > \ln 2 . Certain analytic expressions for M ( a ) M(a) in the complementary range, 0 > a ≤ ln ⁡ 2 0 > a \leq \ln 2 , are also provided.

Apparently new summations in terms of well-known special functions are deduced for hypergeometric-type series containing a digamma function as a factor. As a by-product of this investigation new reduction formulae for the Kampe de Feriet function F0:2;12:1;0[x, x] are obtained.

Using a simple method, numerous summation formulas for hypergeometric and basic hypergeometric series are derived. Among these summation formulas are nonterminating extensions and q-extensions of identities recorded by Lavoie, Luke, Watson, and Srivastava. At the result side of the basic hypergeometric summations, there appears a q-analogue of the digamma function. Some of its properties are also studied.

Recently, interesting novel summation formulae for hypergeometric-type series containing a digamma function as a factor have been established by Miller (2006 J. Phys. A: Math. Gen. 39 3011 - 20) mainly by exploiting already known results and using certain transformation and reduction formulae in the theory of the Kampe de Feriet double hypergeometric function. It is shown here that these, and several other series of this type, can be closed-form summed by simpler and more direct arguments based only on a derivative formula for the Pochhammer symbol and the theory of the digamma ( or.) function and generalized hypergeometric function. In addition, a new reduction formula for the Kamp e de Feriet function F 1:1;0 (1 2;1) [z, z] is obtained.

Let ψ ( x ) denote the digamma function, that is, the logarithmic derivative of Euler's Γ-function. Let q be a positive integer greater than 1 and γ denote Euler's constant. We show that all the numbers ψ ( a / q ) + γ , ( a , q ) = 1 , 1 ⩽ a ⩽ q , are transcendental. We also prove that at most one of the numbers γ , ψ ( a / q ) , ( a , q ) = 1 , 1 ⩽ a ⩽ q , is algebraic.

The goal of this paper is to prove the following asymptotic formula Γ ( x ) ≈ 2 π e − b ( x + b ) x exp ( − x − 1 2 ψ ( x + c ) ) as x ∈ N , x → ∞ , where Γ is the Euler Gamma function and ψ is the digamma function, namely, the logarithmic derivative of Γ . Moreover, optimal values of parameters b , c are calculated in such a way that this asymptotic convergence is the best possible.

The modified k splines are introduced. A simple linear model results from applying a variational principle to a modified 3-spline approximation of Φ′ on [1,2]. An approximation of Φ is obtained analytically by integration. The coefficients needed to use this approximate method are tabulated.

In composite materials, the inclusions are frequently spaced very closely. The electric field concentrated in the narrow regions between two adjacent perfectly conducting inclusions will always become arbitrarily large. In this paper, we establish an asymptotic formula of the electric field in the zone between two spherical inclusions with different radii in three dimensions. An explicit blowup factor relying on radii is obtained, which also involves the digamma function and Euler-Mascheroni constant, and so the role of inclusions' radii played in such blowup analysis is identified.

The sums of three trigonometric series with logarithmic coefficients are derived by extending an approach first utilized by Lerch. By applying Frullani's theorem to two of these series, two non-trivial integrals involving hyperbolic functions are evaluated in terms of the digamma function., Published version

In a previous paper, we showed that the elliptic digamma function, defined by the logarithmic derivative of the elliptic gamma function, satisfies an addition type formula. The integrals appearing in this formula can be considered to be one-parameter deformations of q-double zeta values and thus two-parameter deformations of double zeta values. In this paper, we introduce certain integrals, regarded as two-parameter deformations of multiple zeta values, and investigate their properties. In particular, we consider two-parameter generalizations of the harmonic and shuffle product formulas, which are fundamental relations for multiple zeta values.

We offer two new Mellin transform evaluations for the Riemann zeta function in the region $0, Comment: Corrected some typos

By computing the derivatives of five classical hypergeometric summation theorems, and applying the related properties of the digamma function, we derive a large number of closed summation formulae for generalized harmonic numbers.

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.

In this paper, we propose a numerical method to estimate the unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid. The implicit numerical method is employed to solve the direct problem. For the inverse problem, we first obtain the fractional sensitivity equation by means of the digamma function, and then we propose an efficient numerical method, that is, the Levenberg–Marquardt algorithm based on a fractional derivative, to estimate the unknown order of a Riemann–Liouville fractional derivative. In order to demonstrate the effectiveness of the proposed numerical method, two cases in which the measurement values contain random measurement error or not are considered. The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes’ first problem for a heated generalized second grade fluid. In this paper, we propose a numerical method to estimate the unknown order of a Riemann–Liouville fractional derivative for a fractional Stokes first problem for a heated generalized second grade fluid. The implicit numerical method is employed to solve the direct problem. For the inverse problem, we obtain the fractional sensitivity equation by means of the digamma function. Numerical simulations demonstrate the effectiveness of the proposed method.

In the paper, the author shows that the partial sums are alternatively larger and smaller than the generalized Euler’s harmonic numbers with sharp bounds, where γ is the Euler's constant, are the Bernoulli numbers and ψ is the digamma function.

In this paper, the two-dimensional fractional cable equation is considered, an efficient numerical method to obtain the identification of the fractional derivatives is investigated. Concerning the numerical treatment of the two-dimensional fractional cable equation, a fourth-order compact finite difference method is proposed, the stability and convergence of the compact difference method are discussed rigorously by means of the Fourier method. For the inverse problem of the identification of the fractional derivatives, Levenberg---Marquardt iterative method is employed, and the fractional sensitivity equation is obtained by means of the digamma function. Finally, numerical examples are presented to show the effectiveness of the proposed numerical method.

We present one-parameter series representations for the following series involving the Riemann zeta function ??n=3 n odd ?(n)/n sn and ??n=2 n even ?(n) n sn and we apply our results to obtain new representations for some mathematical constants such as the Euler (or Euler-Mascheroni) constant, the Catalan constant, log 2, ?(3) and ?.

Integral means are important class of bivariate means. In this paper we prove the very general algorithm for calculation of coefficients in asymptotic expansion of integral mean. It is based on explicit solving the equation of the form $B(A(x))=C(x)$, where $B$ and $C$ have known asymptotic expansions. The results are illustrated by calculation of some important integral means connected with gamma and digamma functions.

In this paper, by using the method of partial fraction decomposition and integral representations of series, we establish some expressions of series involving harmonic numbers and binomial coefficients in terms of zeta values and harmonic numbers. Furthermore, we can obtain some closed form representations of sums of products of quadratic (or cubic) harmonic numbers and reciprocal binomial coefficients, and some explicit evaluations are given as applications. The given representations are new.

In this paper, we establish some lower and upper bounds for certain hyperbolic functions in terms of partial sums of their Taylor series. These allow us to present two completely monotonic functions involving gamma function. As consequences, sharp Burnside type approximations for gamma function, sharp Detemple–Wang type approximations for psi function, and sharp bounds for polygamma functions are deduced, which generalize and improve some known results.

In this paper, two classes of completely monotonic functions involving the gamma function are investigated and exploited to establish sharp inequalities for the gamma and the digamma functions. These results extend to the q-gamma function for all \(q>0\). Sharp bounds for the q-gamma and the q-digamma functions are provided in terms of the inverse hyperbolic function (arcsinh).

In this paper the fractional order derivative of a Dirichlet series, Hurwitz zeta function and Riemann zeta function is explicitly computed using the Caputo fractional derivative in the Ortigueira sense. It is observed that the obtained results are a natural generalization of the integer order derivative. Some interesting properties of the fractional derivative of the Riemann zeta function are also investigated to show that there is a chaotic decay to zero (in the Gaussian plane) and a promising expression as a complex power series.

The present paper aims at introducing and investigating a new class of generalized double zeta function i.e. modified double zeta function which involves the Riemann, Hurwitz, Hurwitz-Lerch, Barnes double zeta function and Bin-Saad generalized double zeta function as particular cases. The results are obtained by suitably applying Riemann-Liouville type and Tremblay fractional integral and differential operators. We derive the expansion formula for the proposed function with some of its properties via fractional operators and discuss the link with known results.

By the use of methods of real analysis and weight functions, we study the equivalent properties of a Hilbert-type integral inequality with the nonhomogeneous kernel. The constant factor related to the Hurwitz zeta function is proved to be the best possible. As a corollary, a few equivalent conditions of a Hilbert-type integral inequality with the homogeneous kernel are deduced. We also consider their operator expressions.

In the spirit of Euler sums we develop a set of identities for finite sums of products of harmonic numbers in higher order and reciprocal binomial coefficients. The new results complement some Euler sums of the type

We prove the following very accurate approximation formula for the factorial function: This gives better results than the following approximation formula which is established by the author [5] and C. Mortici [16] independently, and gives similar results with which is established by C. Mortici in his very new paper [8].

The aim of this paper is to introduce some simple and fast formulas for approximating the gamma function. Some involved functions are completely monotonic. The corresponding asymptotic series are constructed and some sharp inequalities are established.

In this paper, we provide an asymptotic formula for the higher derivatives of the Hurwitz zeta function with respect to its first argument that does not need recurrences. As a by-product, we correct some formulas that have appeared in the literature.

In this paper, we mainly give much simple proof to Theorem 1.1 and (1.6) of Theorem 1.2 posed in the paper ?F. Qi, Limit formulas for ratios between derivatives of the gamma and digamma functions at their singularities, Filomat 27 (2013) 601-604.?

The constants of Landau and Lebesgue are defined, for all integers n ? 0 , in order, by G n = ? k = 0 n 1 16 k 2 k k 2 and L n = 1 2 π ? - π π sin n + 1 2 t sin 1 2 t d t , which play important roles in the theories of complex analysis and Fourier series, respectively. Diverse inequalities and approximations for these constants have been investigated and developed by many authors. Here, in this paper, we establish new asymptotic expansions for the constants G n and L n / 2 of Landau and Lebesgue, respectively, in terms of the digamma and polygamma functions. Based on our expansion for the Landau constants G n , we present new bounds for the Landau constants G n in terms of the digamma and polygamma functions. We also establish inequalities for the Lebesgue constants L n / 2 , which are applied to derive an asymptotic expansion for L n / 2 in terms of 1 / ( n + 1 ) . Furthermore, by giving numerical calculations to be compared, among several developed asymptotic expansions for the constants G n and L n / 2 , it is shown that our expansions presented here would be best ones.

In this short note, we point out an interesting connection between series with zeta values and certain infinite products. Using this connection we give a closed-form evaluation of a special class of series with zeta values in the coefficients.

We aim to introduce a $\mathtt{k}$-Struve function and investigate its various properties, including mainly certain inequalities associated this function. One of the inequalities given here is pointed out to be related to the so-called classical Tur\'an type inequality. We also present a differential equation, several recurrence relations, and integral representations for this $\mathtt{k}$-Struve function.

Both the Gamma function Γ(z) and the Psi (or Digamma) function ψ(z) are meromorphically continued to the whole complex z-plane with singularities (which are simple poles) at z=−k for non-positive integer values of k. Here, in this presentation, we derive the following (presumably new) limit formulas: for positive integer values of n and q, k being a non-negative integer. One of the above limit formulas is used here to partition the singularities of Γ(z) into the first set of singularities (at z=0,−1,−2,−3) and the second set of singularities (at z=−4,−5,−6, …). Some other closely-related formulas are also considered.

We consider the transcendence of the multi-indexed series∑n1,…,nk=1∞f(n1,…,nk)n1⋯nk and then extend our results to series of the form∑n1,…,nk=0∞f(n1,…,nk)A1(n1)⋯Ak(nk)B1(n1)⋯Bk(nk) where f is a k-periodic function and each Ai(x),Bi(x) is a polynomial with algebraic coefficients. We relate these series to special values of the digamma function as well as polygamma functions. We also show some conditional implications on the transcendence of the series via Schanuel's Conjecture.

The main result of this article is to establish new and accurate approximations of the gamma function in terms of the digamma function, that improve a result of Alzer and Batir [Appl. Math. Lett. 20(2007):778–781].

The aim of this paper is to construct the asymptotic series of the ratio of gamma functions by Kershaw then to deduce some sharp estimates. Mathematical subject classification: 33B15, 05A16, 26D15.

We give a purely algebraic proof of a formula for Taylor coefficients of the reciprocal gamma function. The formula expresses each coefficient in terms of multiple zeta values. Our proof uses Hoffman’s harmonic algebra of multiple zeta values.