The aim of this paper is to derive some new generalized fractional analogues Mercer type inequalities, essentially using the convexity property functions and Raina’s function. We also discuss several special cases which show that our results are, an extent, unifying. In order illustrate significance results, we offer interesting applications means, error bounds, q-digamma functions.

We discuss the best methods available for computing gamma function Γ(z) in arbitrary-precision arithmetic with rigorous error bounds. address different cases: rational, algebraic, real or complex arguments; large small low high precision; without precomputation. The also cover log-gamma log Γ(z), digamma ψ(z), and derivatives Γ⁽ⁿ⁾(z) ψ⁽ⁿ⁾(z). Besides attempting to summarize existing state of ar...

We utilize the definition of a fractional integral operators, which was presented by Ahmad et al., to investigate general fractional-type identity with parameter. establish certain parameterized inequalities based on this identity, and provide two examples illustrate obtained results. Also, these results derived in paper are applied estimations q-digamma function, divergence measures cumulative...

Abstract Derivatives with respect to the parameters of integral Mittag-Leffler function and Wright function, recently introduced by us, are calculated. These derivatives can be expressed in form infinite sums quotients digamma gamma functions. In some particular cases, these calculated closed-form help MATHEMATICA. However, parameter differentiation reduction formulas explicitly derived order c...

The two-fold aim of the paper is to unify and generalize on the one hand the double integrals of Beukers for ζ(2) and ζ(3), and those of the second author for Euler’s constant γ and its alternating analog ln(4/π), and on the other hand the infinite products of the first author for e, and of the second author for π and e . We obtain new double integral and infinite product representations of man...

In this paper, the neutrix limit is used to extend the definition of the Hurwitz zeta function ζ(α, x) and its partial derivatives to the whole complex plane except for non-positive integers α, in particular, the values of ζ(1, x) is obtained. This definition is equivalent to the Hermite’s integral of ζ(α, x) as α 6= 1, 0,−1, . . .. Moreover, some properties of ζ(1, x) are established and we fi...

In this paper, we establish new generalizations of the Hermite-Hadamard-type inequalities. These inequalities are formulated in terms modules certain powers proper functions. Generalizations for convex functions also considered. As applications, some digamma function trigamma and involving special means real numbers given. The results include estimates via arithmetic, geometric logarithmic mean...

The paper is devoted to the piece-wise analytic case of Meijer's $G$ function $G^{m,n}_{p,p}$. While problem its continuation was solved in principle by Meijer and Braaksma we show that ''balanced'' $m+n=p$ formulas take a particularly simple form. We derive explicit expressions for values these continuations on banks branch cuts. It further demonstrated particular cases this type having intege...

Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states hydrogen atom. In textbooks, eigenvalue are defined for linear problems, particularly differential equations such as time-independent Schr\"odinger equations. Eigenfunctions exhibit several standard features independent form underlying As discussed Bender ...

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...