Harrumph indeed! Simply replacing n!^p /(n+w)!^p by 1/Product[n+k,{k,1,w}] makes it easy: Sum[n!^p/(n+w)!^p,{n,0,oo}] == HypergeometricPFQRegularized[{1,1 ..p+1 copies}, {1 + w, 1 + w,.. p copies}, 1] In[1]:= Table[Sum[1/(Product[n + k, {k, 1, w}])^p, {n, 0, \[Infinity]}] // FullSimplify, {w, 1, 7, 2}, {p, 3, 7, 2}] // InputForm // Timing Out[1]={2.781 Second, { {Zeta[3], Zeta[5], Zeta[7]} , {29/32 - (3*Zeta[3])/ 4, (-3*(-791 + 520*Zeta[3] + 160*Zeta[5]))/ 512, (-3*(-35731 + 21448*Zeta[3] + 8288*Zeta[5] + 1344*Zeta[7]))/ 4096} , {(5*(-2077 + 1728*Zeta[3]))/ 1327104, (-1186062547 + 781704000*Zeta[3] + 237634560*Zeta[5])/ 330225942528, (5*(-1404612185513 + 848813853888*Zeta[3] + 322757084160*Zeta[5] + 49203044352*Zeta[7]))/ 4565043429507072} , {(7*(389467 - 324000*Zeta[3]))/ 503884800000, (-7*(-784046040073 + 516756240000*Zeta[3] + 157075200000*Zeta[5]))/ 125382662553600000000, (156827579578950685957 - 94800261054346848000*Zeta[3] - 36021058152710400000*Zeta[5] - 5475321411840000000*Zeta[7])/584985350410076160000000000000} } } W. -----Original Message----- From: R. William Gosper [mailto:rwg@osots.com] Sent: maandag 30 mei 2005 9:10 To: math-fun@mailman.xmission.com Subject: Re: [math-fun] FW: Beautiful Zeta(3) formulas! rcs>Well, cute anyway. I assume these aren't news to some of our readers. I should hope not.
-----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. 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 =============================== This email is confidential and intended solely for the use of the individual to whom it is addressed. If you are not the intended recipient, be advised that you have received this email in error and that any use, dissemination, forwarding, printing, or copying of this email is strictly prohibited. You are explicitly requested to notify the sender of this email that the intended recipient was not reached.