Notes on generalization of the Bernoulli type polynomials

Recently, Srivastava et al. introduced a new generalization of the Bernoulli, Euler and Genocchi polynomials (see [H.M. Srivastava, M. Garg, S. Choudhary, Russian J. Math. Phys. 17 (2010) 251-261] and [H.M. Srivastava, M. Garg, S. Choudhary, Taiwanese J. Math. 15 (2011) 283-305]). They established several interesting properties of these general polynomials, the generalized Hurwitz-Lerch zeta functions and also in series involving the familiar Gaussian hypergeometric function. By the same motivation of Srivastava’s et al. [11] and [12], we introduce and derive multiplication formula and some identities related to the generalized Bernoulli type polynomials of higher order associated with positive real parameters a, b and c. We also establish multiple alternating sums in terms of these polynomials. Moreover, by differentiating the generating function of these polynomials, we give a interpolation function of these polynomials.     

Further summation formulae related to generalized harmonic numbers

By employing the univariate series expansion of classical hypergeometric series formulae, Shen [L.-C. Shen, Remarks on some integrals and series involving the Stirling numbers and ζ(n), Trans. Amer. Math. Soc. 347 (1995) 1391-1399] and Choi and Srivastava [J. Choi, H.M. Srivastava, Certain classes of infinite series, Monatsh. Math. 127 (1999) 15-25; J. Choi, H.M. Srivastava, Explicit evaluation of Euler and related sums, Ramanujan J. 10 (2005) 51-70] investigated the evaluation of infinite series related to generalized harmonic numbers. More summation formulae have systematically been derived by Chu [W. Chu, Hypergeometric series and the Riemann Zeta function, Acta Arith. 82 (1997) 103-118], who developed fully this approach to the multivariate case. The present paper will explore the hypergeometric series method further and establish numerous summation formulae expressing infinite series related to generalized harmonic numbers in terms of the Riemann Zeta function ζ(m) with m=5,6,7, including several known ones as examples.     

Simultaneous operational calculus involving a product of a general class of polynomials,Fox’sH-function and the multivariableH-function

New operational relations between the original and the image for two-dimensional Laplace transforms involving a general class of polynomials, Fox’sH-function and the multivariableH-function are obtained. The result provides a unification of the bivariate Laplace transforms for theH-functions given by Chaurasia [2, 3].     

Some generalizations of the Apostol-Genocchi polynomials and the Stirling numbers of the second kind

Recently, the authors introduced some generalizations of the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials (see [Q.-M. Luo, H.M. Srivastava, J. Math. Anal. Appl. 308 (2005) 290-302] and [Q.-M. Luo, Taiwanese J. Math. 10 (2006) 917-925]). The main object of this paper is to investigate an analogous generalization of the Genocchi polynomials of higher order, that is, the so-called Apostol-Genocchi polynomials of higher order. For these generalized Apostol-Genocchi polynomials, we establish several elementary properties, provide some explicit relationships with the Apostol-Bernoulli polynomials and the Apostol-Euler polynomials, and derive various explicit series representations in terms of the Gaussian hypergeometric function and the Hurwitz (or generalized) zeta function. We also deduce their special cases and applications which are shown here to lead to the corresponding results for the Genocchi and Euler polynomials of higher order. By introducing an analogue of the Stirling numbers of the second kind, that is, the so-called λ-Stirling numbers of the second kind, we derive some basic properties and formulas and consider some interesting applications to the family of the Apostol type polynomials. Furthermore, we also correct an error in a previous paper [Q.-M. Luo, H.M. Srivastava, Comput. Math. Appl. 51 (2006) 631-642] and pose two open problems on the subject of our investigation.     

Criteria for univalence of the Dziok-Srivastava and the Srivastava-Wright operators in the class A

In the geometric function theory (GFT) much attention is paid to various linear integral operators mapping the class S of the univalent functions and its subclasses into themselves. In [12] and [13] Hohlov obtained sufficient conditions that guarantee such mappings for the operator defined by means of Hadamard product with the Gauss hypergeometric function. In our earlier papers as [20], [19], [17] and [18], etc., we extended his method to the operators of the generalized fractional calculus (GFC, [16]). These operators have product functions of the forms _m+1F_m and _m+1Ψ_m and integral representations by means of the Meijer G- and Fox H-functions. Here we propose sufficient conditions that guarantee mapping of the univalent, respectively of the convex functions, into univalent functions in the case of the celebrated Dziok-Srivastava operator ([8] : J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput.103, No 1 (1999), pp. 1-13) defined as a Hadamard product with an arbitrary generalized hypergeometric function _pF_q. Similar conditions are suggested also for its extension involving the Wright _pΨ_q-function and called the Srivastava-Wright operator (Srivastava, [36]). Since the discussed operators include the above-mentioned GFC operators and many their particular cases (operators of the classical FC), from the results proposed here one can derive univalence criteria for many named operators in the GFT, as the operators of Hohlov, Carlson and Shaffer, Saigo, Libera, Bernardi, Erdélyi-Kober, etc., by giving particular values to the orders p ? q + 1 of the generalized hypergeometric functions and to their parameters.     

On composition of some general fractional integral operators

In the present paper we derive three interesting expressions for the composition of two most general fractional integral oprators whose kernels involve the product of a general class of polynomials and a multivariableH-function. By suitably specializing the coefficients and the parameters in these functions we can get a large number of (new and known) interesting expressions for the composition of fractional integral operators involving classical orthogonal polynomials and simpler special functions (involving one or more variables) which occur rather frequently in problems of mathematical physics. We have mentioned here two special cases of the first composition formula. The first involves product of a general class of polynomials and the Fox’sH-functions and is of interest in itself. The findings of Buschman [1] and Erdélyi [4] follow as simple special cases of this composition formula. The second special case involves product of the Jacobi polynomials, the Hermite polynomials and the product of two multivariableH-functions. The present study unifies and extends a large number of results lying scattered in the lierature. Its findings are general and deep.     

On unified fractional integral operators

The present paper is in continuation to our recent paper [6] in these proceedings. Therein, three composition formulae for a general class of fractional integral operators had been established. In this paper, we develop the Mellin transforms and their inversions, the Mellin convolutions, the associated Parseval-Goldstein theorem and the images of the multivariableH-function together with applications for these operators. In all, seven theorems and two corollaries (involving the Konhauser biorthogonal polynomials and the Jacobi polynomials) have been established in this paper. On account of the most general nature of the polynomials S _n ^m [x] and the multivariableH-function whose product form the kernels of our operators, a large number of (new and known) interesting results involving simpler polynomials and special functions (involving one or more variables) obtained by several authors and hitherto lying scattered in the literature follow as special cases of our findings. We give here exact references to the results (in essence) of seven research papers which follow as simple special cases of our theorems.     

Multiple integrals involving theH-function of two variables

In this paper, we establish two multiple integrals involving the products of generalised hypergeometric function,H-function of one and two variables. The integrals evaluated are very general in nature and generalise the well-known integrals due to Mittal and Gupta, Goyal, Olkha, Dahiya and others. Some interesting integrals involving the products of elementary special functions and orthogonal polynomials have also been obtained as particular cases of the main results.     

A theorem concerning a product of two general classes of polynomials and the multivariableH-function

A theorem concerning a product of two general classes of polynomials and the multivariableH-function is established. Certain integrals and expansion formulae have also been derived by the application of this theorem. This general theorem yields a number of new, interesting and useful theorems, integrals and expansion formulae as its particular cases.     

Extended Srivastava’s triple hypergeometric <Emphasis Type="Italic">H</Emphasis><Subscript><Emphasis Type="Italic">A,p,q</Emphasis></Subscript> function and related bounding inequalities

In this paper, motivated by certain recent extensions of the Euler’s beta, Gauss’ hypergeometric and confluent hypergeometric functions (see [4]), we extend the Srivastava’s triple hypergeometric function H _A by making use of two additional parameters in the integrand. Systematic investigation of its properties including, among others, various integral representations of Euler and Laplace type, Mellin transforms, Laguerre polynomial representation, transformation formulas and a recurrence relation, is presented. Also, by virtue of Luke’s bounds for hypergeometric functions and various bounds upon the Bessel functions appearing in the kernels of the newly established integral representations, we deduce a set of bounding inequalities for the extended Srivastava’s triple hypergeometric function H _A,p,q .     

Fractional integral formulae involving a general class of polynomials and the multivariableH-function

We obtain two fractional integral formulae involving a general class of polynomials and the multivariableH-function. On account of the most general nature of the polynomials and the multivaribleH-function involved herein, our findings provide interesting unifications and extensions of a number of (known and new) results. We have mentioned here only two such results.     

Convolution integral equations involving a general class of polynomials and the multivariableH-function

In this paper we first solve a convolution integral equation involving product of the general class of polynomials and theH-function of several variables. Due to general nature of the general class of polynomials and theH-function of several variables which occur as kernels in our main convolution integral equation, we can obtain from it solutions of a large number of convolution integral equations involving products of several useful polynomials and special functions as its special cases. We record here only one such special case which involves the product of general class of polynomials and Appell's functionF ₃. We also give exact references of two results recently obtained by Srivastavaet al [10] and Rashmi Jain [3] which follow as special cases of our main result.     

A note on the generalization of the summation formulas for Appell's and Kampé de Fériet's hypergeometric functions

In the present paper, a summation formula of a general triple hypergeometric series F⁽³⁾(x, y, z) introduced by Srivastava [10] is obtained. A particular case of this formula corresponds to a result of Shah [7] involving Kampé de Fériet's double hypergeometric function which can further be specialized to yield summation formulas of Srivastava [11] and Bhatt [2] for Appell's function F₂.     

On some applications of the Briot-Bouquet differential subordination

Recently Srivastava et al. [J. Dziok, H.M. Srivastava, Certain subclasses of analytic functions associated with the generalized hypergeometric function, Integral Transforms Spec. Funct. 14 (2003) 7-18; J. Dziok, H.M. Srivastava, Classes of analytic functions associated with the generalized hypergeometric function, Appl. Math. Comput. 103 (1999) 1-13; Y.C. Kim, H.M. Srivastava, Fractional integral and other linear operators associated with the Gaussian hypergeometric function, Complex Var. Theory Appl. 34 (1997) 293-312] introduced and studied a class of analytic functions associated with the generalized hypergeometric function. In the present paper, by using the Briot-Bouquet differential subordination, new results in this class are obtained.     

Certain fractional derivative formulae involving the product of a general class of polynomials and the multivariableH-function

In the present paper, we obtain three unified fractional derivative formulae (FDF). The first involves the product of a general class of polynomials and the multivariableH-function. The second involves the product of a general class of polynomials and two multivariableH-functions and has been obtained with the help of the generalized Leibniz rule for fractional derivatives. The last FDF also involves the product of a general class of polynomials and the multivariableH-function but it is obtained by the application of the first FDF twice and it involves two independent variables instead of one. The polynomials and the functions involved in all our fractional derivative formulae as well as their arguments which are of the typex ^ρ Π _i=1 ^s (x ^t _i +α _i ) ^σ _i are quite general in nature. These formulae, besides being of very general character have been put in a compact form avoiding the occurrence of infinite series and thus making them useful in applications. Our findings provide interesting unifications and extensions of a number of (new and known) results. For the sake of illustration, we give here exact references to the results (in essence) of five research papers [2, 3,10, 12, 13] that follow as particular cases of our findings. In the end, we record a new fractional derivative formula involving the product of the Hermite polynomials, the Laguerre polynomials and the product ofr different Whittaker functions as a simple special case of our first formula.     

On properties of hypergeometric functions of three variables,F M andF G

The behavior of Lauricella hypergeometric seriesF _M andF _G (see [2]) near the boundary points of their domains of convergence is discussed. Such properties for one variable series, the Gauss₂ F ₁ and the Clausen₃ F ₂, and for two variables, the AppellF ₁,F ₂ andF ₃, are established by the author [5], [7] and the results are effectively applied to solve problems for the Euler-Darboux equation and to calculate multiplications of the fractional calculus operators in the articles by the author [4], [6] and by H. M. Srivastava and the author [9].     

Integrals involving Fox’sH-function

We evaluate four integrals involving Fox’sH-functions and a general class of polynomialsS _n ^m [x], introduced earlier by Srivastava.     

New families of generating functions for certain class of three-variable polynomials

Recently, Srivastava, Özarslan and Kaanoglu have introduced certain families of three and two variable polynomials, which include Lagrange and Lagrange-Hermite polynomials, and obtained families of two-sided linear generating functions between these families [H.M. Srivastava, M.A. Özarslan, C. Kaanoglu, Some families of generating functions for a certain class of three-variable polynomials, Integr. Transform. Spec. Funct. iFirst (2010) 1-12]. The main object of this investigation is to obtain new two-sided linear generating functions between these families by applying certain hypergeometric transformations. Furthermore, more general families of bilinear, bilateral, multilateral finite series relationships and generating functions are presented for them.     

Some families of generating functions for the extended Srivastava polynomials

Almost four decades ago, H.M. Srivastava considered a general family of univariate polynomials, the Srivastava polynomials, and initiated a systematic investigation for this family [10]. In 2001, B. González, J. Matera and H.M. Srivastava extended the Srivastava polynomials by inserting one more parameter [4]. In this study we obtain a family of linear generating functions for these extended polynomials. Some illustrative results including Jacobi, Laguerre and Bessel polynomials are also presented. Furthermore, mixed multilateral and multilinear generating functions are derived for these polynomials.     

Addition theorems for the Appell polynomials and the associated classes of polynomial expansions

Various interesting and potentially useful properties and relationships involving the Bernoulli, Euler and Genocchi polynomials have been investigated in the literature rather extensively. Recently, the present authors (Srivastava and Pinter in Appl Math Lett 17:375–380, 2004) obtained addition theorems and other relationships involving the generalized Bernoulli polynomials ${B_n^{(\alpha)}(x)}$ and the generalized Euler polynomials ${E_n^{(\alpha)}(x)}$ of order α and degree n in x. The main purpose of this sequel to some of the aforecited investigations is to give several addition formulas for a general class of Appell sequences. The addition formulas, which are derived in this paper, involve not only the generalized Bernoulli polynomials ${B_n^{(\alpha)}(x)}$ and the generalized Euler polynomials ${E_n^{(\alpha)}(x)}$ , but also the generalized Genocchi polynomials ${G_n^{(\alpha)}(x)}$ , the Srivastava polynomials ${\mathcal{S}_{n}^{N}\left( x\right)}$ , several general families of hypergeometric polynomials and such orthogonal polynomials as the Jacobi, Laguerre and Hermite polynomials. Some umbral-calculus generalizations of the addition formulas are also investigated.