Identities On Special Functions

The main results of this thesis are some new methods for proving and deducing special function identities including automated proofs of identities on Bernoulli and Euler polynomials based on the extended Zeilberger's algorithm, the q-operator proofs of formulae on the bivariate Rogers-Szeg(o|¨) polynomials which are closely related to the classical orthogonal polynomials, the functional-theoretic method to verify theta function identities and the application of theta decompositions to deduce new theta function identities.The first chapter is devoted to a background knowledge on the main methods to prove special function identities, including Zeilberger's algorithm and its extensions, q-difference operators, contiguous relations and theta functions. We also present some basic definitions and notations that are used throughout the thesis.In Chapter 2, we present a computer algebra approach to prove identities on Bernoulli polynomials and Euler polynomials by using the extended Zeilberger's algorithm given by Paule. The key idea is to use the contour integral definitions of the Bernoulli and Euler numbers to transform the identities on these numbers and polynomials into integral identities on hypergeometric integrands. The recurrence relations established by the extended Zeilberger's algorithm for the integrands have certain parameter free properties which lead to the recurrence relations of the required identities without computing the integrals. Furthermore, new identities on Bernoulli numbers are derived.In Chapter 3, we present an operator approach to deriving Mehler's formula and the Rogers formula for the bivariate Rogers-Szeg(o|¨) polynomials h_n(x,y|q). Our proof of Mehler's formula can be considered as a new approach to verify the nonsymmetric Poisson kernel formula for the continuous big q-Hermite polynomials H_n(x; a|q) due to Askey, Rahman and Suslov. Mehler's formula for h_n(x, y|q) involves a 3φ2 sum and the Rogers formula involves a 2φ1 sum. The proofs of these results are based on the parameter augmentation with respect to the q-exponential operator and the homogeneous q-shift operator in two variables. By extending the known results on the Rogers-Szeg(o|¨) polynomials h_n(x|q), we obtain another Rogers-type formula for h_n(x,y|q). Finally, we give a change of base formula for H_n(x; a|q) which can be used to evaluate some integrals by using the Askey-Wilson integral.In Chapter 4, we generalize the functional-theoretic method for proving Jacobi's triple product identity to verify theta function identities by employing the contiguous relations and the initial coefficients. The two-term contiguous relations satisfied by a given theta function can be obtained by straightforward calculations and the number of the initial values depends on the dimension of the theta function. As applications, several well-known identities are proved, such as the quintuple product identity, the septuple product identity and Riemann's addition formula. Furthermore, we introduce the notion of theta decompositions to derive new theta function identities. We explain how to transform the set of the theta function solutions of a given contiguous relation into a finite dimensional vector space over the field K(q). The new theta function identities can be derived by considering the linear dependence relations among the elements of the vector space.