Contributions To Computer Proofs Of Q-Hypergeometric Identities

The subject of computer proofs of identities began with the Ph. D thesis [17] of Sister Mary Celine Fasenmyer at the University of Michigan in 1945. Then in 1982 Zeilberger [49] realized that Sister Celine's method for obtaining pure recurrence relations satisfied by hy-pergeometric polynomials opened the door to automatic proving hypergeometric identities. At the beginning of 1990s, based on Gosper's algorithm, Zeilberger [51,52] gave a creative telescoping algorithm (also called Zeilberger's algorithm) for fast finding recurrence relations that are satisfied by hypergeometric sums, which has made possible a whole generation of computerized proofs of identities.When Zeilberger studied Sister Celine's Method in the late 1970s, he realized that one can prove a hypergeometric identity by checking a finite number of special cases. In the book "A=B" [37, p. 70], it is pointed that it is an interesting research question to ask how small we can make this a priori estimate of the number. In this thesis, we focus on this question for q-hypergeometric identities.In 1993, Yen in her doctoral dissertation [45] gave the first a priori estimate of the number for hypergeometric identities, it is extremely large. Furthermore, in 1996, Yen [47] gave an estimate of the number n₁ satisfying that for q-hypergeometric identitiesif the equality holds for all n ∈ {n₀,n₀+1, …,n₁} then∑_kF(n,k) =1 for all n≥n₀, as a polynomial of degree 24 in the parameters of F(n, k).The method to estimate the number n\ is proving that the two sides of a q-hypergeometric identity satisfy the same recurrence, and giving both an estimate of the order J of the recurrence and an estimate of the integer m₁ such that the leading coefficient of the recurrence is not identically zero, when n ≥ m₁. Therefore we have n₁=max{J,m₁}.We first use Sister Celine's method to prove the q-analog [43, Theorem 5.1] of the fundamental theorem [37, Theorem 4.4.1], i.e., every q-proper hypergeometric term satisfies a k-free recurrence. From this theorem, we prove the existence of the recurrence for the admissible q-proper hypergeometric sums, and obtain an estimate of the order J of the recurrence. Then we present three methods to estimate the number n₁.Firstly, we prove a proposition: the leading coefficient (regarded as a polynomial in qⁿ, q^1/2) of the recurrence is not equal to zero for all n ≥ [m/2] + 1, where m is the degree in q^1/2 of the leading coefficient. Then we use Cramer's rule to obtain an estimate of the degree in q^1/2) of the leading coefficient of the recurrence. Consequently, we get an estimate formula of the number n₁, bounded above by a polynomial of degree 24 in the parameters of F(n, k). For instance, we get n₁ = 16666 for the q-Vandermonde-Chu identity.Secondly, we generalize Sister Celine's method, using the existence of the solutions of the homogeneous linear equations to obtain an estimate of the degree in q^1/2) of theleading coefficient of the recurrence. Then we get an estimate formula of the number i\;bounded above by a polynomial of degree 9 in the parameters of F(n;k). As pointed by D. Zeilberger;this is vastly improving the important previous bounds of L. Yen. For instance;we get ?i = 1741 for the ^-Vandermonde-Chu identity.Finally;by combining Wu's elimination method and the generalization of Sister Celine's method;we give a fast elementary algorithm for #-hypergeometric identitiesk) = G(n); n>n0to obtain a smaller estimate of the number ti\. At the same time;we implement the algorithm in Maple program. For instance;we get ni = 191 for the ^-Vandermonde-Chu identity.