[math-fun] FW: Beautiful Zeta(3) formulas!
Well, cute anyway. I assume these aren't news to some of our readers. Rich -----Original Message----- From: Number Theory List [mailto:NMBRTHRY@LISTSERV.NODAK.EDU] On Behalf Of Troy Kessler Sent: Friday, May 27, 2005 6:46 AM To: NMBRTHRY@LISTSERV.NODAK.EDU Subject: Beautiful Zeta(3) formulas! 29/32 -3*Zeta[3]/4 = sum n=0 to oo (n)!^3/(n+3)!^3 (mathematica can do) 5*(-2077+1728Zeta[3])/1327104 = sum n=0 to oo (n)!^3/(n+5)!^3 Mathematica gives an answer that is 5 lines long for this second sum. It can be shown that if 3 or 5 is replaced with a odd number the result is rational + rational *Zeta[3]. (Use Zeilberger in A=B)
rcs>Well, cute anyway. I assume these aren't news to some of our readers. I should hope not. My 1974 MIT AI Lab paper Acceleration of Series (completed by Guy Steele when I absquatulated to California) derived a recurence(k) for Sum(n!^3/(n+2k+1)!^3,n>1). These are all rational (zero bits/term) summands, but when k goes oo, nothing remains but the unbeautiful "orphaned" sum, inf 2 80 k + 29 k 3 ==== (k + ---------) (- 1) k! \ 56 (d175) 14 > --------------------------- / 2 ==== (2 k + 1) (3 k + 3)! k = 0 with the redeeming convergence rate of three trits/term: (c176) dfloat(zeta(3) = apply_nouns(subst(12,inf,%))) Time= 151 msec. (d176) 1.20205690315959d0 = 1.20205690315959d0
-----Original Message----- From: Number Theory List [mailto:NMBRTHRY@LISTSERV.NODAK.EDU] On Behalf Of Troy Kessler Sent: Friday, May 27, 2005 6:46 AM To: NMBRTHRY@LISTSERV.NODAK.EDU Subject: Beautiful Zeta(3) formulas!
29/32 -3*Zeta[3]/4 = sum n=0 to oo (n)!^3/(n+3)!^3 (mathematica can do) 5*(-2077+1728Zeta[3])/1327104 = sum n=0 to oo (n)!^3/(n+5)!^3 Mathematica gives an answer that is 5 lines long for this second sum. It can be shown that if 3 or 5 is replaced with a odd number the result is
rational + rational *Zeta[3]. (Use Zeilberger in A=B)
Harrumph. (c154) closedform(sum(n!^3/(n+5)!^3,n,1,inf)) Time= 1131 msec. 5 zeta(3) 1298221 (d154) --------- - --------- 768 165888000 (c156) closedform(sum(n!^3/(n+7)!^3,n,1,inf)) Time= 2163 msec. 935108917 7 zeta(3) (d156) --------------- - --------- 172832486400000 1555200 Not to mention things like (c194) closedform(sum(n!*(n+1)!*(n+3)!/(n+7)!/(n+6)!/(n+4)!,n,1,inf))) Time= 931 msec. 3635021 zeta(3) (d194) --------- - ------- 435456000 144 I can give (c197) sum(n!*(n+a)!*(n+b)!/(n+2*k+a+1)!/(n+2*k+2*a+1)!/(n+2*k-b+2*a+1)!,n,1,inf) (d197) inf ==== \ n! (n + a)! (n + b)!
---------------------------------------------------------------- / (n + 2 k + a + 1)! (n + 2 k + 2 a + 1)! (n + 2 k - b + 2 a + 1)! ==== n = 1
explicitly as zeta(3) [1 0] M1(0) M1(1)...M1(a-1) M2(0)...M2(b-1) M3(0)...M3(k-1) [ ], 1 where the Ms are diagonal 2x2 matrices with (cumbersome) rational function entries. Furthermore, I can give inf ==== \ n! (n + a)! (n + b)!
--------------------------------------, 0<=a<=b<=c<=d<=e, / (n + c + 1)! (n + d + 1)! (n + e + 1)! ==== n = 1
explicitly as [zeta(2)] [0 1 0] M1(0)...M1(b-a) M2(0)...M6(e-d) [zeta(3)] with M1...M6 [ 1 ] 3x3 matrices, and sums of n! (n + a)! (n + b)! (n + c)!/...(n + g)! as analogous products of 4x4s times a vector containing zeta(2),...,zeta(4), etc, ad memoriam virtuum. --rwg
participants (2)
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R. William Gosper -
Schroeppel, Richard