Oh foo, I was almost there, but stopped too soon. The lhs was a bilateral sum. The rhs was the limit of a 2F1[1/(1-e^(i t))] that I linearly transformed and didn't notice that it became the "1H1" bilateral sum: Sum[(E^(I n t) (a+n)!)/(b+n)!,{n,-Infinity,Infinity}]== - E^(-(1/2) I (1+a+b) (-\[Pi]+t)) \[Pi] Csc[a \[Pi]] (2 Sin[t/2])^(-1-a+b)/(-1-a+b)! I doubt this is written to hold as widely as possible. I got it by q->1 in Ramanujan's 1 psi 1. Where should I have looked it up instead? --rwg On Mon, Mar 14, 2011 at 7:02 PM, Bill Gosper <billgosper@gmail.com> wrote:

quad>

I saw this sum somewhere as a question as to whether it has a closed form. I am pretty sure it doesn't have any elementary closed form, but I'd think that this isn't a whole new class of functions (with respect

to special functions).

(%i1) 'sum(1/(x^k + k!), k, 0, inf); inf ==== \ 1 (%o1) > -------

/ k ==== x + k! k = 0

for some real x (we might even just consider positive integral x). Maybe even just

(%i2) 'sum(1/(1 + k!), k, 0, inf); inf ==== \ 1 (%o2) > ------

/ k! + 1 ==== k = 0

Can we write this in any other useful forms? (cue the Gosperfunctionology)

-Robert -----------------------

You gotta be kidding. I'm trying to show the kids bilateral infinite matrix products and I can't even "shew that"

Limit[(E^(I N t) (-b + N)! Hypergeometric2F1[1, a - N, b - N, E^(-I t)])/(-a + N)!,

N -> \[Infinity]] == -(2^(-a + b) E^(1/2 I (b (-3 Pi + t) + a (Pi + t))) Pi Csc[b Pi]/ ((-Sin[t/2])^(a - b) (-1 + E^(I t)) Gamma[-a + b]))

(integer N, b>a, t>pi/2, and probably more widely).

Where's the nearest home for the bewildered? --rwg