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Integrals that contain polylogarithmic functions in the integrand are occasionally represented in closed form by utilizing specialized Euler sums. We investigate a specific family of polylogarithmic integrals and demonstrate its reduction to Euler sums, which in certain instances can be expressed in terms of special functions such as the Riemann zeta function. This research extends the findings of Freitas and provides a solution to a question posed by Chavan, Kobayashi, and Layja.

The introduction of two-parameter (p,q){(p,q)}-calculus and Lie algebras in 1991 has spurred a wave of recent research into (p,q){(p,q)}-special polynomials, including (p,q){(p,q)}-Bernoulli, (p,q){(p,q)}-Euler, (p,q){(p,q)}-Genocchi and (p,q){(p,q)}-Frobenius–Euler polynomials. These investigations have been carried out by numerous researchers in order to uncover a wide range of identities associated with these polynomials and applications. In this article, we aim to introduce (p,q){(p,q)}-sine and (p,q){(p,q)}-cosine based λ-array type polynomials and derive numerous properties of these polynomials such as (p,q){(p,q)}-integral representations, (p,q){(p,q)}-partial derivative formulae and (p,q){(p,q)}-addition formulae. It is worth noting that the utilization of the (p,q){(p,q)}-polynomials introduced in this study, along with other (p,q){(p,q)}-polynomials, can lead to the derivation of various identities that differ from the ones presented here.

Various extensions of the beta function together with their associated extended hypergeometric and confluent hypergeometric functions have been introduced and investigated. In this paper, using the very recently contrived extended beta function, we aim to introduce an extension Fvp,q;λ;σ,τu{{}_{u}F_{v}^{p,q;\lambda;\sigma,\tau}}of the generalized hypergeometric function Fvu{{}_{u}F_{v}}and investigate certain classes of transforms and several identities of a generalized probability distribution involving this extension. In fact, we present some interesting formulas of Jacobi, Gegenbauer, pathway, Laplace, and Legendre transforms of this extension multiplied by a polynomial. We also introduce a generalized probability distribution to investigate its several related properties. Further, we consider some special cases of our main results with an argument about the derived process of a known result.

Motivated mainly by certain interesting recent extensions of the generalized hypergeometric function [H. M. Srivastava, A. Çetinkaya and I. Onur Kıymaz, A certain generalized Pochhammer symbol and its applications to hypergeometric functions, Appl. Math. Comput. 226 2014, 484–491] by means of the generalized Pochhammer symbol, we introduce here a new extension of the generalized Mittag-Leffler function. We then systematically investigate several properties of the extended Mittag-Leffler function including some basic properties, Mellin, Euler-Beta, Laplace and Whittaker transforms. Furthermore, certain properties of the Riemann–Liouville fractional integrals and derivatives associated with the extended Mittag-Leffler function are also investigated. Some interesting special cases of our main results are pointed out.

Since the so-called Hermite–Hadamard type inequalities for convex functions were presented, their generalizations, refinements, and variants involving various integral operators have been extensively investigated. Here we aim to establish several Hermite–Hadamard–Fejér type inequalities for symmetrized convex functions and Wright-quasi-convex functions with a weighted function symmetric with respect to the midpoint axis on the interval involving the known generalized fractional integral operators. We also point out that certain known inequalities are particular cases of the results presented here.

A variety of polynomials, their extensions and variants have been extensively investigated, due mainly to their potential of applications in diverse research areas. In this sequel, we introduce a new class of generalized Laguerre–Euler polynomials and present certain potentially useful formulas and identities such as implicit summation formulae and symmetry identities. The new class of polynomials introduced and the results presented here, being very general, are shown to be specialized to reduce to various known polynomials and to yield some known and new formulas and identities, respectively.

Raina (East Asian Math J 21(2):191–203, 2005) introduced a new fractional integral operator which is a generalization of Riemann–Liouville fractional integral. Agarwal et al. (Fasc Math 204:5–27, 2016) provided some Ostrowski type fractional integral inequalities. Chen (J Math Inequal 10(1):75–81, 2016) gave extension and refinement of the Hermite–Hadamard inequality for convex functions via Riemann–Liouville fractional integrals. Here, motivated by the above-mentioned works, we aim at establishing extension and refinement of the Hermite–Hadamard type inequalities for a function with certain conditions by using new fractional integral operators introduced by Raina and Agarwal et al. above. The inequalities presented here are also pointed out to include some known results, as their special cases.

In this paper, we aim to introduce Legendre-based poly-Euler polynomials and Legendre-based Multi Poly-Euler polynomials and investigate some related identities and formulas such as addition formulas, implicit summation formula, and symmetry identities. We also point out that the results presented here, being very general, can be specialized to give many known and new identities and formulas involving relatively simple numbers and polynomials.

We establish a result for the product of two operators defined on a Lie algebra of endomorphisms of a vector space. Then we use this result to derive some properties for Gegenbauer polynomials, for example, Rodrigues formula. The method developed here is potentially useful to investigate some other special functions of mathematical physics.

In this paper, we choose to investigate some interesting and useful identities found in the ‘Lost’ notebook of Ramanujan and give their generalizations by using the well-known q-binomial theorem. We also consider some special cases of our main results presented here, some of which are indicated to have relevant connections with those involving relatively simpler and earlier identities.

AbstractWe present a method to produce estimations of the natural logarithmic constant e, accurate to as many decimal places as we desire. The method is based on an asymptotic formula for (1 + 1/x)x, which uses the partition function.

AbstractWe first present the corrected expression for a certain widely-recorded generalized Goldbach–Euler series. The corrected forms are then shown to be connected with the problem of closed-form evaluation of series involving the Zeta functions, which happens to be an extensively-investigated subject since the time of Euler as (for example) in the classical three-century-old Goldbach theorem.

Among recent investigations of fractional integral operators including various extensions of the Mittag-Leffler function in the kernel, very recently, a very generalized fractional integral operator containing a further extension of the Mittag-Leffler function has been introduced and investigated. In this paper, we aim to establish some new generalized Chebyshev type inequalities involving this generalized fractional integral operator. Relevant connections of the results presented here with those involving relatively simple fractional integral operators are also pointed out.

Ever since the time of Euler, the so-called Euler sums have been evaluated in many different ways. We give here a (presumably) new proof of the classical Euler sum. We show that several interesting analogues of the Euler sums can be evaluated by systematically analyzing some known summation formulas involving hypergeometric series. Many other identities related to the Euler sums are also presented.

Among a remarkably large number of various extensions of polynomials and numbers, and diverse introductions of new polynomials and numbers, in this paper, we choose to introduce two new generalizations of some extended Bernoulli polynomials and numbers by using the Mittag–Leffler function and the confluent hypergeometric function. Then, we investigate certain properties and formulas of these newly introduced polynomials and numbers such as explicit representations, addition formulas, integral formulas, differential formulas, inequalities, and inequalities involving their integrals. Also, by using the theory of umbral calculus, five additional formulas regarding these new polynomials are provided. Furthermore, we propose to introduce four generalizations of the extended Euler and Genocchi polynomials. Finally, three natural problems are poised.

The authors apply the theory of the double gamma function, which was recently revived in the study of determinants of Laplacians, to evaluate some families of series involving the Riemann zeta function. Introducing a (presumably new) mathematical constant in the theory of the double gamma function, they also systematically evaluate a definite integral of the double gamma function and various classes of series associated with zeta functions. Some of these definite integrals are expressed in terms of quotients of double gamma functions.

We evaluate the sums of certain classes of series involving the Riemann zeta function by using the theory of the double gamma function, which has recently been revived in the study of determinants of Laplacians. Relevant connections with various known results are also pointed out.

Abstract.:  The authors evaluate some interesting families of infinite series by analyzing known identities involving generalized hypergeometric series. Several special cases of the main results are shown to be related to earlier works on the subject.

Very recently Srivastava et al. (Russ J Math Phys 25(1):116–138, 2018) have introduced the incomplete H-functions and investigated their several interesting properties, for example, decomposition and reduction formulas, derivative formulas, and various integral transforms. They also pointed out potential applications of many of those incomplete special functions, which are specialized from the incomplete H-functions, involving (for example) probability theory. In this paper, we aim to establish two pathway fractional integral formulas involving the incomplete H-functions. Also our main results are indicated to reduce to yield some known identities. Further, among numerous special cases of our main results, some of them are explicitly demonstrated.