Date: Fri, 16 Dec 2005 13:37:17 -0800 (PST) From: "R. William Gosper" <rwg@osots.com> Subject: [math-fun] more hyjynx [...] My POWERSERIES function got a double sum instead of Clausen's 4F3[z] for not knowing the well-poised, 3-balanced (sesquiSaalschutzian?), terminating 4F3[1] in the inner sum. "SesquiPfaffian" is more just, but also more confusing. I got ambitious and derived the whole 4F3[a,b,c,d;e,f,g] (and 3F2[z]) matrix system, and extracted the general, noterminating case: 1 hyper_f ([a, b, c, c - b - a + -], 4, 3 2 1 [c - a + 1, c - b + 1, b + a + -], 1) = 2 1 (b + a - -)! (c - a)! (c - b)! (sqrt(%pi) (c - 2 b - 2 a)! 2 1 1 /((a - -)! (b - -)! (c - 2 a)! (c - 2 b)! (c - b - a)!) 2 2 - hyper_f ([1, c - 2 a + 1, c - 2 b + 1, c - b - a + 1], 4, 3 3 [c + 1, c - 2 b - 2 a + 2, c - b - a + -], 1) 2 c + 1 1 /((a - 1)! (b - 1)! (----- - b - a) c! (c - b - a + -)!)) 2 2 Notice the rhs 4F3 getting divided by infinity exactly when an upper parameter on the left is a nonpositive integer. Also note the weird oo - oo when (c+1)/2 = a+b. As usual, the matrices provide the complete system of identities contiguous to the above. [...] ------------------ I.e., termination of lhs provides a closed form. The lhs remains finite for c=2a+2b-1, so it must be that block([fancy_display:false],hyper_f[3,2]([b,a,((b+a)/2)],[b+a,((b+a+1)/2)],1), print(%%=hypersimp(%%)),0)$ b + a b + a + 1 hyper_f ([b, a, -----], [b + a, ---------], 1) = 3, 2 2 2 2 b + a + 1 %pi gamma (---------) 2 --------------------------- 2 a + 1 2 b + 1 gamma (-----) gamma (-----) 2 2 which I found startling because it's balanced like zeta(3/2) and nonterminating and square. But as you see, Macsyma already knows it, so I simply forgot installing Clausen's theorem. This was a specialization of 4x4 matrices to 3x3s. The old message used "3 balanced" to mean like zeta(3), in violation of the convention that n-balanced means like zeta(n+1), in disregard of that consarned invisible lower parameter "1". We can probably get away with calling zeta(2)-like "Pfaffian", since the completely unrelated standard usage is as a noun. --rwg