This is a continuation of [19], where we presented an extension of the q-hypergeometric function with connection to the title of this paper. In chapter one we present some quadratic q − hypergeometric transformations, to give more examples of this extension. In chapter two, systems of partial q-difference equations for the q-Appell and q-Lauricella functions are presented in the authors notation. Other attempts to find these equations were made by Jackson. It turns out that these q-difference equations can be written in many equivalent forms, which gives rise to the notion of equivalence class for q-difference equations. In chapter three q-analogues of expansion formulas by Chaundy [11] and Burchnall & Chaundy [9] are found. In the process we obtain a corrected version of [26, (55) p.79]. In chapter four we find an expan-sion formula for a Φ 2:1 2:0 function by using Jackson’s sum of a terminating very-well-poised balanced 8φ7 series. We also find the corresponding q-binomial identity.

We proceed with pseudo-q-Appell polynomials in the spirit of [12]. It turns out that these q-Bernoulli numbers are the same as BJHC,ν,q. As in [12] we find q-analogues of many formulas in [38], the umbral calculus works remarkably well also for pseudo-q-Appell pol., only the q is put up instead of down corresponding to inversion of basis. We also find new q-Euler-Maclaurin expansions.

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We first study four different types of q-Hermite polynomials hν,q(x), ψν,q(x), kν,q(x), fν,q(x) from the point of view of generating functions and operational formulas. Three of these q-Hermite polynomials are q-Appell polynomials, and the remaining one is a pseudo q-Appell polynomial. q-difference equations and q-hypergeometric expressions are found. In this paper, we only obtain Rodriguez formulas for two of the four polynomials, namely hν,q(x) and kν,q(x), and accordingly we can only find q-analogues of the so-called Nielsen’s formula for these two polynomials. Matrix forms for the polynomials expressed by q-Pascal matrices are considered. An orthogonality relation for kν,q(x) expressed as a q-integral is found. The two Cigler q-Laguerre polynomials are introduced and with the help of the relationship between hν,q(x) and the q-Laguerre polynomials, new generating functions for hν,q(x) are found. In the second part of the paper we study orthogonality relations for q-Jacobi-, q-Laguerre- and q-Legendre polynomials. The proofs will all use q-integration by parts, a method equivalent to the previously used recurrence technique. The orthogonality relations are all of discrete type.

We consider the Γq function for 0 < |q | < 1 and complex function values. q-Analogues of Euler’s constant, the Gaussian Ψ function, the Euler and Weierstrass formulas for Γ(z) are introduced. The meromorphic continuation of the Γqfunction is found. For the q-Riemann zeta function [26], we show a multiplication formula with the Γq function. The Jacobi elliptic functions sn u, cn u and dn u may be expressed in the form sin x, cosx and 1 times a balanced Γq function. We give a solution of the Truesdell [40] Fq equation.

The quantum groups and Hopf algebras have invaded theoretical physics. We give an alternate (and possibly more understandable) description of these phenomena in this paper. In another paper [9] the author has introduced a q-umbral calculus in the spirit of Rota. This umbral calculus contains two dual q–additions, the Nalli–Ward–Alsalam (NWA) q–addition and the Jackson–Hahn–Cigler (JHC) q–addition. In a further paper [10], a matrix calculus was built up from the Jackson q-exponential function. It is well-known that Lie groups (one-parameter subgroups) can be made up from the exponential function applied to the Lie algebra (e.g. Pauli matrices). A combination of these two techniques will lead to a new and natural q-deformation of matrix Lie groups. We will use two matrix multiplications, the first one is ordinary multiplication, and the second one contains an involution operator τ: q → 1 q. We supply these two matrix multiplications with an attractive associative structure. It turns out that these deformed matrix pseudo–groups have quite a different character than the well-known quantum groups. In spite of this, we have decided to keep the same notation for these q-deformed matrix groups as for the corresponding quantum groups. The first matrix pseudo-group to be defined SOq(2), is not seen so often explicitly in the quantum group context. However several physicists use the corresponding SOq(3) symmetric quantum mechanics [11], [30].

Abstract not found