A method for q-calculus

We present a notation for q-calculus, which leads to a new method for computations and classifications of q-special functions. With this notation many formulas of q-calculus become very natural, and the q-analogues of many orthogonal polynomials and functions assume a very pleasant form reminding directly of their classical counterparts. The first main topic of the method is the tilde operator, which is an involution operating on the parameters in a q-hypergeometric series. The second topic is the q-addition, which consists of the Ward-AlSalam q-addition invented by Ward 1936 [102, p. 256] and Al-Salam 1959 [5, p. 240], and the Hahn q-addition. In contrast to the the Ward-AlSalam q-addition, the Hahn q-addition, compare [57, p. 362] is neither commutative nor associative, but on the other hand, it can be written as a finite product. We will use the generating function technique by Rainville [76] to prove recurrences for q-Laguerre polynomials, which are q-analogues of results in [76]. We will also find q-analogues of Carlitz' [26] operator expression for Laguerre polynomials. The notation for Cigler's [37] operational calculus will be used when needed. As an application, q-analogues of bilinear generating formulas for Laguerre polynomials of Chatterjea [33, p. 57], [32, p. 88] will be found.