TWO CURIOUS q-ANALOGUES OF HERMITE POLYNOMIALS
نویسندگان
چکیده
Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider two new q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci polynomials and the recent works on the combinatorics of the Matrix Ansatz of the PASEP.
منابع مشابه
A curious q-analogue of Hermite polynomials
Two well-known q-Hermite polynomials are the continuous and discrete q-Hermite polynomials. In this paper we consider a new family of q-Hermite polynomials and prove several curious properties about these polynomials. One striking property is the connection with q-Fibonacci and q-Lucas polynomials. The latter relation yields a generalization of the Touchard-Riordan formula.
متن کاملClasses of Bivariate Orthogonal Polynomials
We introduce a class of orthogonal polynomials in two variables which generalizes the disc polynomials and the 2-D Hermite polynomials. We identify certain interesting members of this class including a one variable generalization of the 2-D Hermite polynomials and a two variable extension of the Zernike or disc polynomials. We also give q-analogues of all these extensions. In each case in addit...
متن کاملOn a Reduction Formula for a Kind of Double q-Integrals
Abstract: Using the q-integral representation of Sears’ nonterminating extension of the q-Saalschütz summation, we derive a reduction formula for a kind of double q-integrals. This reduction formula is used to derive a curious double q-integral formula, and also allows us to prove a general q-beta integral formula including the Askey–Wilson integral formula as a special case. Using this double ...
متن کاملThe Cauchy Operator for Basic Hypergeometric Series
We introduce the Cauchy augmentation operator for basic hypergeometric series. Heine’s 2φ1 transformation formula and Sears’ 3φ2 transformation formula can be easily obtained by the symmetric property of some parameters in operator identities. The Cauchy operator involves two parameters, and it can be considered as a generalization of the operator T (bDq). Using this operator, we obtain extensi...
متن کاملVariants of the Rogers-ramanujan Identities
We evaluate several integrals involving generating functions of continuous q-Hermite polynomials in two diierent ways. The resulting identities give new proofs and generalizations of the Rogers-Ramanujan identities. Two quintic transformations are given, one of which immediately proves the Rogers-Ramanujan identities without the Jacobi triple product identity. Similar techniques lead to new tra...
متن کامل