Vagner Jikia, Ilia Lomidze
We have shown that in some region where the Euler integral of the first kind diverges, the Euler formula defines a generalized function. The connected of this generalized function with the Dirac delta function is found.
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In the paper, the author elementarily unifies and generalizes eight identities involving the functions $\frac{\pm1}{e^{\pm t}-1}$ and their derivatives. By one of these identities, the author establishes two explicit formulae for computing Euler polynomials and two-parameter Euler polynomials, which are a newly introduced notion, in terms of Stirling numbers of the second kind.
The asymptotic expansions of the Wright functions of the second kind, introduced by Mainardi [see Appendix F of his book {\it Fractional Calculus and Waves in Linear Viscoelasticity}, (2010)], $$ F_σ(x)=\sum_{n=0}^\infty \frac{(-x)^n}{n! \g(-nσ)}~,\quad M_σ(x)=\sum_{n=0}^\infty \frac{(-x)^n}{n! \g(-nσ+1-σ)}\quad(0<σ<1)$$ for $x\to\pm\infty$ are presented. The situation corresponding to the limit $σ\to1^-$ is considered, where $M_σ(x)$ approaches the Dirac delta function $δ(x-1)$. Numerical results are given to demonstrate the accuracy of the expansions derived in the paper, together with graph…
Let $\un,$ be a quasi-definite linear functional defined on the space of polynomials $\mathbb{P}.$ For such a functional we can define a sequence of monic orthogonal polynomials (SMOP in short) $(P_n)_{n\geq 0},$ which satisfies a three term recurrence relation. Shifting one unity the recurrence coefficient indices given the sequence of associated polynomials of the first kind which are orthogonal with respect to a linear functional denoted by $\un^{(1)}$. In the literature two special transformations of the functional $\un$ are studied, the canonical Christoffel transformation $\widetilde \un…
Using the direct relation between the Gegenbauer polynomials and the Ferrers function of the first kind, we compute interrelations between certain Jacobi polynomials, Meixner polynomials, and the Ferrers function of the first kind. We then compute Rodrigues-type and orthogonality relations for Ferrers functions of the first and second kinds. In the remainder of the paper using the relation between Gegenbauer polynomials and the Ferrers function of the first kind we derive connection and linearization relations, some definite integral and series expansions, some asymptotic expansions of Mehler-…
The classical form of Grüss' inequality was first published by G. Grüss in 1935 and gives an estimate of the difference between the integral of the product and the product of the integrals of two functions. After that many variants of this inequality appeared in the literature. The aim of this paper is to consider some Chebyshev-Grüss-type inequalities and apply them to Bernstein-Euler-Jacobi (BEJ) operators of first and second kind. First and second moments of the operators are used to explain the situation.
In the present article, we study Bell based Euler polynomial of order α and investigate some useful correlation formula, summation formula and derivative formula. Also, we introduce some relation of string number of the second kind. Moreover, we drive several important formulae of bell based Euler polynomial by using umbral calculus.
Five series are evaluated in terms of zeta values. Three of the series involve harmonic numbers and one involves Stirling numbers of the first kind. The evaluation of these series is reduced to the evaluation of certain integrals, including the moments of the polylogarithm.
The aim of this paper is to construct generating functions for new families of special polynomials including the Appel polynomials, the Hermite-Kampè de Fèriet polynomials, the Milne-Thomson type polynomials, parametric kinds of Apostol type numbers and polynomials. Using Euler's formula, relations among special functions, Hermite-type polynomials, the Chebyshev polynomials and the Dickson polynomials are given. Using generating functions and their functional equations, various formulas and identities are given. With help of computational formula for new families of special polynomials, some o…
In the present paper we introduce some expansions, based on the falling factorials, for the Euler Gamma function and the Riemann Zeta function. In the proofs we use the Faá di Bruno formula, Bell polynomials, potential polynomials, Mittag-Leffler polynomials, derivative polynomials and special numbers (Eulerian numbers and Stirling numbers of both kinds). We investigate the rate of convergence of the series and give some numerical examples.
The definition for the $p$-adic Hurwitz-type Euler zeta functions has been given by using the fermionic $p$-adic integral on $\mathbb Z_p$. By computing the values of this kind of $p$-adic zeta function at negative integers, we show that it interpolates the Euler polynomials $p$-adically. Many properties are provided for the $p$-adic Hurwitz-type Euler zeta functions, including the convergent Laurent series expansion, the distribution formula, the functional equation, the reflection formula, the derivative formula, the $p$-adic Raabe formula and so on. The definition for the $p$-adic Euler $L$…
The hyperharmonic numbers h_{n}^{(r)} are defined by means of the classical harmonic numbers. We show that the Euler-type sums with hyperharmonic numbers: σ(r,m)=\sum_{n=1}^{\infty}((h_{n}^{(r)})/(n^{m})) can be expressed in terms of series of Hurwitz zeta function values. This is a generalization of a result of Mező and Dil. We also provide an explicit evaluation of σ(r,m) in a closed form in terms of zeta values and Stirling numbers of the first kind. Furthermore, we evaluate several other series involving hyperharmonic numbers.
An orthogonality relation for the Whittaker functions of the second kind of imaginary order, $W_{κ,\mathrm{i}μ}(x)$, with $μ\in\mathbb{R}$, is investigated. The integral $\int_{0}^{\infty}\mathrm{d}x\: x^{-2}W_{κ,\mathrm{i}μ}(x)W_{κ,\mathrm{i}μ'}(x)$ is shown to be proportional to the sum $δ(μ-μ')+δ(μ+μ')$, where $δ(μ\pmμ')$ is the Dirac delta distribution. The proportionality factor is found to be $π^{2}/[μ\sinh(2πμ)Γ({1/2}-κ+\mathrm{i}μ) Γ({1/2}-κ-\mathrm{i}μ)]$. For $κ=0$ the derived formula reduces to the orthogonality relation for the Macdonald functions of imaginary order, discussed rece…
We propose a systematic study of transformations of $A$-hypergeometric functions. Our approach is to apply changes of variables corresponding to automorphisms of toric rings, to Euler-type integral representations of $A$-hypergeometric functions. We show that all linear $A$-hypergeometric transformations arise from symmetries of the corresponding polytope. As an application of the techniques developed here, we show that the Appell function $F_4$ does not admit a certain kind of Euler-type integral representation.
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