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For a class of even weight functions, generalizations of the classical Jacobi and Laguerre polynomials are defined via fractional Rodrigues' type formulas using the Bessel operators in the form ( d 2 / d x 2) + ((2 β + 1) / x) (d / d x). Their properties, including hypergeometric representation, differential recurrence relations and fractional boundary value problem, are investigated. [ABSTRACT FROM AUTHOR]

Recently, Casadei [4] provided an explicit formula for statistical marginal model in terms of Poisson-Gamma mixture. This model involving certain polynomials which play the key role in reference analysis of the signal and background model in counting experiments. The principal object of this paper is to present a natural further step toward the mathematical properties concerning this polynomials. We Ąrst obtain explicit representations for these polynomials in form of the Laguerre polynomials and the conĆuent hyper-geometric function and then based on these representations we derive a number of useful properties including generating functions, recurrence relations, differential equation, Rodrigues formula, Ąnite sums and integral transforms. [ABSTRACT FROM AUTHOR]

In this manuscript, we present the generalized hypergeometric function of the type F r r , r > 2 and extension of the K Laguerre polynomial for the K extended Laguerre polynomials A r , n , k α x . Additionally, we describe the K generating function, K recurrence relations, and K S Rodrigues formula. [ABSTRACT FROM AUTHOR]

New sequences of orthogonal polynomials with ultra-exponential weight functions are discovered. In particular, we give an explicit solution to the Ditkin–Prudnikov problem (1966). The 3-term recurrence relations, explicit representations, generating functions and Rodrigues-type formulae are derived. The method is based on differential properties of the involved special functions and their representations in terms of the Mellin–Barnes and Laplace integrals. A notion of the composition polynomial orthogonality is introduced. The corresponding advantages of this orthogonality to discover new sequences of polynomials and their relations to the corresponding multiple orthogonal polynomial ensembles are shown. [ABSTRACT FROM AUTHOR]