Discussions On Some Problems From The WZ Theory

This thesis is a preliminary survey which is concerned with two interesting features of the WZ theory due to H.Wilf and D. Zeilberge. One fold is the Gosper equation closely related to Gosper algorithm, another is identities regarding the WZ equation.In Chapter one, we focus on the so–called Gosper equation a(n)x(n + 1)- b(n- 1)x(n) = c(n).A new sum formula is given. Especially with the help of Mathematica software, we discuss in detail the Gosper equation provided the coefficient polynomials a(n), b(n), c(n)are of degree 5, wherein some new and interesting results are found.In Chapter two, we introduce the inverse of the usual difference operator when redefined as to be an action on the matrix algebras. Using them, we generalize [10,Theorem7.1.1] to the following k-1i=0F(n, i)- F(0, i)=n-1i=0G(i, k)- G(i, 0),∞j=k F(n, j)- f j=∞i=n G(i, k)- g i),where(F, G) is an arbitrary WZ pair subject to the so–called WZ equation F(n + 1, k)- F(n, k) = G(n, k + 1)- G(n, k),f j= lim nâ†'∞F(n, j), g i= lim kâ†'∞G(i, k).Furthermore, from such manner some new combinatorial identities associated with WZ–pairs are derived.In Chapter three, we deduce from the WZ equation F(n + 1, k)- F(n, k) = G(n, k + 1)- G(n, k),the finite forms of three infinite identities,among includes n-1k=0G(k, 0) =n-1k=0F(k + 1, k) + G(k, k)-n-1k=0F(n, k).By analogy to Amdeberhan and Zeilberger’s accelerative convergendce method, we apply the obtained to evaluate the famous Zata function ζ(s), s = 2, 3.