Some Applications Of Operational Methods On Special Function

In this paper, we study some applications of the operational methods on special function, for example, we use the operational methods to generalize some polynomial functions, introduce some new special functions, also we use the op-erational methods to investigate some properties of q-series. In accordance with section order, the abstract are as follows:(1) We use the operational methods to generalize some polynomial func-tions. Firstly, general classes of Tricomi and Hermite-Tricomi functions are in-troduced by exploiting properties of an iterated isomorphism, related to the so-called Laguerre-type exponentials. And we mainly consider the properties of the general classes of 3-variable 2-index Tricomi function and 2-index 4-variable 1-parameter Hermite-Tricomi function, including their generating functions, recur-rence relations, differential equations satisfied by them and some other functional relations, which generalize the main results of S. Khan [80]; Secondly, by using operational methods, we introduce Hermite-based Sheffer polynomials associated with r-variable Gould-Hopper polynomials. As special case, we mainly study the Hermite-based Appell polynomials associated with r-variable Gould-Hopper poly-nomials. Some properties of them are considered, including generating functions and differential equations satisfied by them, which proved to be useful for the derivation of identities involving these polynomials which generalize the results of S. Khan [82].(2) We use the operational methods to introduce some new special functions. Firstly, we use the integral transforms to deal with fractional derivatives. And a new family of special polynomials are introduced. We also study the properties of them, including generating functions, differential equations satisfied by them and some operational relations. And some generalized forms are also considered; Secondly, we introduce a family of hybrid polynomials exhibiting a nature lying between the Laguerre and Legendre polynomials and study their properties by means of appropriate operational rules.(3) We use the operational methods to investigate some properties of the as-sociated Bernoulli polynomials and Euler polynomials. Firstly, we investigate the recurrence relation, the differential equations and connection problems of the 1-variable associated Bernoulli polynomials and Euler polynomials. We generalize them to Hermite-based and investigate their some properties and relation equa-tions, which extend the corresponding results of M. X. He and P. E. Ricci[78], Q.-M. Luo and H. M. Srivastava[110]; Secondly, multivariable Bernoulli polynomi-als are defined generalizing the corresponding generating function of 2-variable Bernoulli polynomials. By using the operational methods, their relevant recur-rence relations, the differential equations satisfied by them and some other prop-erties are obtained. Neo-Bernoulli polynomials are also introduced from a new operational view. And we study some properties of them, which extend the main results of G. Bretti and P. E. Ricci[19].(4) We use the operational methods to investigate some properties of q-series. Firstly, by employing a linear operator, q-Appell-based polynomials are introduced. Some of their properties are considered, including generating func-tions and the differential equations satisfied by them, which proved to be useful for the derivation of relations involving these polynomials; Secondly, we verify the Cauchy operator identities by a new method. And by using the Cauchy operator identities, we obtain a generating function relation for the bivariate Rogers-Szego polynomials. Applying the technique of parameter augmentation to two multiple generalizations of g-Chu-Vandermonde summation theorem given by Milne, we also obtain two multiple generalizations of the Kalnins-Miller transformation.