From: israel@math.ubc.ca (Robert Israel) Subject: Re: More on pi^2 / 6 Date: 28 May 1999 06:22:21 GMT Newsgroups: sci.math Keywords: Generalizations of zeta: Polygamma function, Hurwitz zeta, etc In article <19990527214931.11358.00004585@ng-fv1.aol.com>, Main Night wrote: >It is of course well known that > >1/1 + 1/4 + 1/9 + 1/16 + ... = pi^2 / 6. This makes us wonder, what is > >1/(1.001)^2 + 1/(2.001)^2 + 1/(3.001)^2 + ..... ? 1/(1+x)^2 + 1/(2+x)^2 + 1/(3+x)^2 + ... = Psi(1,1+x) in Maple's notation. This is the "first polygamma function". It is the second derivative of ln(Gamma(x)). In particular, Psi(1,1.001) is approximately 1.6425331958689780329775029123704682185000113199949. > but the question now arises, what is L such that > lim[a-->oo]x(a)=L ? Rather obviously, 0. Note that Psi(1,n+x) = Psi(1,x) - sum_{i=1}^{n-1} 1/(i+x). Robert Israel israel@math.ubc.ca Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2 ============================================================================== From: Robin Chapman Subject: Re: More on pi^2 / 6 Date: Fri, 28 May 1999 07:45:01 GMT Newsgroups: sci.math In article <19990527214931.11358.00004585@ng-fv1.aol.com>, mainnight@aol.com (Main Night) wrote: > (Note: Be warned that the following post has no real non-trivial discoveries in > it, I simply wrote it to get my own thoughts on track. It does have problems > though that might be worthy of attack by those more skilled than myself; so > read on at your own risk, and don't expect any enlightenment.) > > It is of course well known that > > 1/1 + 1/4 + 1/9 + 1/16 + ... = pi^2 / 6. This makes us wonder, what is > > 1/(1.001)^2 + 1/(2.001)^2 + 1/(3.001)^2 + ..... ? > > Indeed, for real a which isnt a negative natural, what is > x=1/(1+a)^2 + 1/(2+a)^2 + 1/(3+a)^2 + ..... ? The Hurwitz zeta function is defined as zeta(s, a) = 1/a^s + 1/(a+1)^s + ... + 1/(a+n)^s + ... There is little loss of generality to suppose that 0 < a<= 1. This converges for Re(s) > 1, and like the Riemann zeta function (the special case a = 1) extends to an analytic function on C save for a simple pole at s = 1. For details see e.g., Whittaker and Watson. For rational a, one can express zeta(s, a) in terms of Dirichlet L-functions. In some cases one can get zeta(2, a) in closed form using these. For a Dirichlet character chi one can get L(2, chi) in closed form whenever chi is even: i.e. chi(-1) = 1. I suspect in most cases one would need odd chi as well to compute zeta(2, a) :-( Robin Chapman + "They did not have proper School of Mathematical Sciences - palms at home in Exeter." University of Exeter, EX4 4QE, UK + rjc@maths.exeter.ac.uk - Peter Carey, http://www.maths.ex.ac.uk/~rjc/rjc.html + Oscar and Lucinda Sent via Deja.com http://www.deja.com/ Share what you know. Learn what you don't. ============================================================================== From: "G. A. Edgar" Subject: Re: More on pi^2 / 6 Date: Fri, 28 May 1999 08:51:56 -0400 Newsgroups: sci.math Another nice generalization is this... infinity ----- \ 1 Pi coth(Pi a) 1 ) ------- = 1/2 ------------- - 1/2 ---- / 2 2 a 2 ----- n + a a n = 1 -- Gerald A. Edgar edgar@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax) ============================================================================== From: "G. A. Edgar" Subject: Re: More on pi^2 / 6 Date: Fri, 28 May 1999 08:54:22 -0400 Newsgroups: sci.math Don't forget this one: infinity ----- \ 1 2 2 2 ) -------- = Pi + Pi cot(Pi a) / 2 ----- (n + a) n = -infinity -- Gerald A. Edgar edgar@math.ohio-state.edu Department of Mathematics telephone: 614-292-0395 (Office) The Ohio State University 614-292-4975 (Math. Dept.) Columbus, OH 43210 614-292-1479 (Dept. Fax) ============================================================================== From: Jim Ferry Subject: Re: More on pi^2 / 6 Date: Fri, 28 May 1999 11:16:02 -0500 Newsgroups: sci.math (This is more to parallel Gerald Edgar's posts than to respond to Robin Chapman's.) Here is an arsenal of formulae for these kinds of problems. The following sums all run from n = -infinity to infinity. ---- \ x pi tan(pi x) = ) --------------- / 2 2 ---- (n + 1/2) - x ---- \ -x pi cot(pi x) = ) ------- / 2 2 ---- n - x ---- n \ (-1) (n + 1/2) pi sec(pi x) = ) --------------- / 2 2 ---- (n + 1/2) - x ---- n+1 \ (-1) x pi csc(pi x) = ) --------- / 2 2 ---- n - x ---- 2 2 \ 1 pi sec (pi x) = ) -------------- / 2 ---- (n + 1/2 + x) ---- 2 2 \ 1 pi csc (pi x) = ) -------- / 2 ---- (n + x) ---- \ x pi tanh(pi x) = ) --------------- / 2 2 ---- (n + 1/2) + x ---- \ x pi coth(pi x) = ) ------- / 2 2 ---- n + x ---- n \ (-1) (n + 1/2) pi sech(pi x) = ) --------------- / 2 2 ---- (n + 1/2) + x ---- n \ (-1) x pi csch(pi x) = ) ------- / 2 2 ---- n + x ---- 2 2 2 2 \ (n + 1/2) - x pi sech (pi x) = ) ------------------ / 2 2 2 ---- ((n + 1/2) + x ) ---- 2 2 2 2 \ x - n pi csch (pi x) = ) --------- / 2 2 2 ---- (n + x ) Whew! | Jim Ferry | Center for Simulation | +------------------------------------+ of Advanced Rockets | | http://www.uiuc.edu/ph/www/jferry/ +------------------------+ | jferry@expunge_this_field.uiuc.edu | University of Illinois |