Multibasic and Mixed Gosper's Algorithm

Gosper s summation algorithm nds a hypergeometric closed form of an inde nite sum of hyperge ometric terms if such a closed form exists We generalize his algorithm to the case when the terms are simultaneously hypergeometric and multibasic q hypergeometric We also provide algorithms for computing hypergeometric canonical forms of rational functions and for nding polynomial solutions of recurrences in the multibasic and mixed case Introduction and notation Let F be a eld of characteristic zero and htni n a sequence of elements from F which is eventually non zero Call tn hypergeometric if there are polynomials p p F x such that p n tn p n tn for all n q hypergeometric or basic hypergeometric if there are polynomials p p F x such that p q n tn p q n tn for all n where q F is a constant called the base multibasic hypergeometric if there are polynomials p p F y ym such that p q n q n m tn p q n q n m tn for all n where q qm F are constants called the bases multibasic and mixed hypergeometric mmHS for short if there are polynomials p p F x y ym such that p n q n q n m tn p n q n q n m tn for all n The celebrated Gosper s algorithm nds hypergeometric solutions fn of the inhomogeneous rst order recurrence fn fn tn where tn is a given hypergeometric sequence Besides its obvious use for inde nite hypergeometric summation it also plays a crucial role in the algorithms for de nite hypergeometric summation con struction of annihilating recurrences and automated veri cation of identities Therefore it is not surprising that analogous algorithms have been designed for many other settings e g integration of hyperexponential functions basic and bibasic hypergeometric summation We present in Section an analogue of Gosper s algorithm for the multibasic and mixed hypergeomet ric case Our algorithm m m Gosper is a common generalization of algorithms presented in Sections and give the required algebraic and algorithmic preliminaries while in Section we develop the multibasic and mixed hypergeometric canonical form of rational functions Although in Gosper s corresponding author available as GosperSum in Mathematica package gosper m at http www cis upenn edu wilf AeqB html algorithm only rst order recurrences are checked for polynomial solutions we provide in Section al gorithm m m Poly which nds all polynomial solutions of inhomogeneous parametric multibasic and mixed recurrences with polynomial coe cients The set of integers is denoted by Z the set of nonnegative integers by N and the eld of rational numbers by Q If n m N and a a a am b b b bm are m tuples of elements of a ring we write ab for the componentwise product a b a b ambm and a n for the componentwise power a a n a n m If m N then we write a for the power product a a a m m We say that two multivariate polynomials over a eld are coprime if they do not have a common non constant factor When a and b are coprime we write a b When S is a set of polynomials and a b for all b S we write a S Algebraic preliminaries Let F be a eld of characteristic zero Let q qm F n f g and suppose that for any integers k km Z q q k qm m k k km This seems to be the right generalization of the condition that q is not a root of unity in the q hypergeometric case see For example if F R q p and q p then q q and we should have chosen q p in the rst place We denote q q qm Let y y y ym be an m tuple of variables F x y the ring of polynomials over F in x and y and F x y the corresponding rational function eld We de ne a di erence operator E on F x y by stipulating that E be xed on F that Ex x and that Eyk qkyk for k m Note that E is inversive that F x y is a di erence eld and that F x y is a di erence subring of F x y see for terminology Let M be the set of power products in y y ym M fy y ym m ki N for i mg If u y y k ym m M we write u q for the corresponding power product of the bases q q qm m Note that Eu u q u for all u M As a multiplicative monoid M is obviously isomorphic to N the direct product of m copies of the additive monoid N We denote by an admissible term order in N which is a total order satisfying for all N An example of an admissible term order is the lexicographic order lex with lex when and k k where k minfi i ig De nition Let N Then we write whenever i i for all i between and m Clearly N is a partial order isomorphic to M j where j denotes divisibility of power products and is contained in any admissible term order for all N We adjoin to N an absorbing bottom element such that for all N De nition Let p F x y Write