Yan-Ping Mu

Applicability of the q-analogue of Zeilberger's algorithm

The applicability or terminating condition for the ordinary case of Zeilbergers algorithm was recently obtained by Abramov. For the q-analogue, the question of whether a bivariate q-hypergeometric term has a qZ-pair remains open. Le has found a solution to this problem when the given bivariate q-hypergeometric term is a rational function in certain powers of q. We solve the problem for the general case by giving a characterization of bivariate q-hypergeometric terms for which the q-analogue of Zeilbergers algorithm terminates. Moreover, we give an algorithm to determine whether a bivariate q-hypergeometric term has a qZ-pair.

Continued Fractions for Rogers–Szegö Polynomials

We evaluate different Hankel determinants of Rogers–Szegö polynomials, and deduce from it continued fraction expansions for the generating function of RS polynomials. We also give an explicit expression of the orthogonal polynomials associated to moments equal to RS polynomials, and a decomposition of the Hankel form with RS polynomials as coefficients.

Recurrent sequences and Schur functions

We show that some classical determinants in the theory of symmetric functions can be interpreted in terms of recurrent sequences. Conversely we generalize determinantal expressions of Schur functions, by taking several recurrent sequences having same characteristic polynomial, or by prolonging sequences to negative indices. Finally, we give some recurrent sequences associated to plethysm of symmetric functions, for example with characteristic polynomial having roots the same powers of the roots of the original characteristic polynomial.

The extended Zeilberger algorithm with parameters

Two hypergeometric terms f(k) and g(k) are said to be similar if the ratio f(k)/g(k) is a rational function of k. For similar hypergeometric terms f1(k),?,fm(k), we present an algorithm, called the extended Zeilberger algorithm, for deriving a linear relation among the sums Fi=?kfi(k)(1?i?m) with polynomial coefficients. When the summands f1(k),?,fm(k) contain a parameter x, we further impose the condition that the coefficients of Fi in the linear relation are x-free. Such linear relations with x-free coefficients can be used to determine the structure relations for orthogonal polynomials and to derive recurrence relations for the connection coefficients between two sequences of orthogonal polynomials. The extended Zeilberger algorithm can be easily adapted to basic hypergeometric terms. As examples, we use the algorithm or its q-analogue to establish linear relations among orthogonal polynomials and to derive recurrence relations with multiple parameters for hypergeometric sums or basic hypergeometric sums.

NON-TERMINATING BASIC HYPERGEOMETRIC SERIES AND THE q-ZEILBERGER ALGORITHM

We present a systematic method for proving non-terminating basic hypergeometric identi- ties. Assume that k is the summation index. By setting a parameter x to xqn, we may find a recurrence relation of the summation by using the q-Zeilberger algorithm. This method applies to almost all non- terminating basic hypergeometric summation formulae in the work of Gasper and Rahman. Furthermore, by comparing the recursions and the limit values, we may verify many classical transformation formu- lae, including the Sears-Carlitz transformation, transformations of the very well-poised 8φ7 series, the Rogers-Fine identity and the limiting case of Watsons formula that implies the Rogers-Ramanujan identities.

Minimal universal denominators for linear difference equations

We provide minimal universal denominators for linear difference equations with fixed leading and trailing coefficients. In the case of first-order equations, they are factors of Abramovs universal denominators. While in the case of higher order equations, we show that Abramovs universal denominators are minimal.

Minimal universal denominators for systems of linear recurrences

Abramov’s algorithm provides universal denominators for rational solutions to the system of linear recurrences of the form . We show that in general Abramov’s estimation is optimal. Meanwhile, we show that better estimations can be obtained when A(x) is a triangular matrix and when A(x) is a constant matrix up to a scalar factor.

Hypergeometric series solutions of linear operator equations

Let K be a field and L:K[x]->K[x] be a linear operator acting on the ring of polynomials in x over the field K. We provide a method to find a suitable basis {bk(x)} of K[x] and a hypergeometric term ck such that y(x)=@?k=0^~ckbk(x) is a formal series solution to the equation L(y(x))=0. This method is applied to construct hypergeometric representations of orthogonal polynomials from the differential/difference equations or recurrence relations they satisfied. Both the ordinary cases and the q-cases are considered.

Parameter augmentation and the q-Gosper algorithm

We develop a method for deriving new basic hypergeometric identities from old ones by parameter augmentation. The main idea is to introduce a new parameter and use the q-Gosper algorithm to find out a suitable form of the summand. By this method, we recover some classical formulas on basic hypergeometric series and find extensions of the Rogers-Fine identity and Ramanujans @j11 summation formula. Moreover, we derive an identity for a @j33 summation.