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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. [ABSTRACT FROM AUTHOR]

Brychkov [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167] evaluated some interesting classes of definite and indefinite integrals involving various special functions and the logarithmic function. Motivated essentially by Brychkov's work [Brychkov YuA. Evaluation of some classes of definite and indefinite integrals. Integral Transforms Spec Funct. 2002;13:163–167], we aim at presenting several classes of definite integrals involving hypergeometric functions and the logarithmic functions, which are expressed explicitly in terms of Psi and generalized zeta functions. Some interesting special cases of our main formulas are given. Relevant connections of the results presented here with those earlier ones are also pointed out. [ABSTRACT FROM AUTHOR]

Computing the matrix elements of the linear operator, which transforms the spherical basis ofSO(3, 1)-representation space into the hyperbolic basis, we present an integral formula involving the product of two Legendre functions of the first kind expressed in terms of4F3-hypergeometric function. Also, using the general Mehler–Fock transform, we obtain another integral formula for the Legendre function of the first kind. A relevant connection of one of the results presented here with a known integral formula is also pointed out. [ABSTRACT FROM PUBLISHER]

Two interesting infinite product formulas were presented by Choiet al. [A generalization of Wilf's formula, Kodai Math. J. 26 (2003), pp. 44–48], who generalized an infinite product formula posed by Wilf as a problem inAmerican Mathematical Monthly[H.S. Wilf,Problem 10588, problems and solutions, Amer. Math. Monthly 104 (1997), p. 456]. Here, we aim mainly at establishing two infinite product formulas, expressed in terms of the Gamma function, whose special cases contain the above generalized infinite product formulas. Our formulas presented here are themselves interesting and their another advantage is the production, easily, of various known or (presumably) new formulas as their special cases. [ABSTRACT FROM AUTHOR]

Abstract: There is an abundant literature on inequalities for the Gamma function Γ and its various related functions as well as their approximations. Only very recently, several authors began to investigate various inequalities for the double Gamma function Γ 2 and its approximation. Here, in this sequel to some of these recent works, we aim at presenting an integral representation of the triple Gamma function Γ 3, which is then used to derive an asymptotic formula for Γ 3. As a by-product of the results presented here, integral representations and asymptotic formulas for the Gamma function Γ and the double Gamma function Γ 2 are also given. The methods and techniques used in this paper can easily be extended to derive the corresponding integral representations and asymptotic formulas for the multiple Gamma functions Γ n (n ≧4). [Copyright &y& Elsevier]

Abstract: Harmonic numbers and generalized harmonic numbers have been studied since the distant past and are involved in a wide range of diverse fields such as analysis of algorithms in computer science, various branches of number theory, elementary particle physics and theoretical physics. Here we aim at presenting further interesting identities about certain finite or infinite series involving harmonic numbers and generalized harmonic numbers. We also give a brief eclectic review on some known properties associated with the harmonic and generalized harmonic numbers. [Copyright &y& Elsevier]

Motivated essentially by their potential for applications in a wide range of mathematical and physical problems, the log-sine integrals have been evaluated, in the existing literature on the subject, in many different ways. The main object of this paper is to present explicit evaluations of some families of log-sine integrals by making use of a familiar algorithm, and also those of log-cosine and other related integrals by using some known integral formulas expressed in terms of the Gamma functions. [ABSTRACT FROM AUTHOR]

Carlson [Some extensions of Lardner's relations between 0F3 and Bessel functions, SIAM J. Math. Anal. 1(2) (1970), 232-242] presented certain connections between Bessel and generalized hypergeometric functions, which generalizes some earlier results. Here, by simply splitting the hypergeometric Appell series F3 into four parts, we show how some useful and generalized relations between F3 and Kampé de Fériet function F0;4;4 2;1;1 can be obtained. These main results are shown to be specialized to yield certain relations between functions 0F1, 0F3, F0;4;4 2;1;1 , and the Humbert function ψ2 some of which are still reduced to produce the Carlson's relations and some other interesting relations between the exponential function, the hyperbolic functions, and modified Bessel functions. Furthermore, decomposition formulas and integral representations of Euler type with hypergeometric function in the kernel for the function F0;4;4 2;1;1 are derived by means of Burchnall-Chaundy operator method which has recently been revived. [ABSTRACT FROM AUTHOR]

Since the time of Euler, the so-called Euler-Mascheroni constant γ has been involved in a wide variety of important mathematical formulas and results. Among these formulas and results, a considerably large number of integral representations of γ have been developed. Here we aim at investigating several further integral expressions for the Euler-Mascheroni constant γ. [ABSTRACT FROM AUTHOR]

We present generalizations of three classical summation formulas 2 F 1 due to Kummer, which are able to be derived from six known summation formulas of those types. As certain simple particular cases of the summation formulas provided here, we give a number of interesting formulas for double-finite series involving quotients of Gamma functions. We also consider several other applications of these formulas. Certain symmetries occur often in mathematical formulae and identities, both explicitly and implicitly. As an example, as mentioned in Remark 1, evident symmetries are naturally implicated in the treatment of generalized hypergeometric series. [ABSTRACT FROM AUTHOR]

Many extensions and variants of the so-called Apostol-type polynomials have recently been investigated. Motivated mainly by those works and their usefulness, we aim to introduce a new class of Apostol-type Laguerre–Genocchi polynomials associated with the modified Milne–Thomson's polynomials introduced by Derre and Simsek and investigate its properties, including, for example, various implicit formulas and symmetric identities in a systematic manner. The new family of polynomials introduced here, being very general, contains, as its special cases, many known polynomials. So the properties and identities presented here reduce to yield those results of the corresponding known polynomials. [ABSTRACT FROM AUTHOR]