A Simple Expression?

Date: 1/26/96 at 8:32:18
From: Anonymous
Subject: Math Problem

Hi -

I heard that you are the right point of contact for any small or big 
math problem.

My problem is the following: I know that 

         infinity  
           sum    a^k/k! = exp(a)
           k=0

But is it possible to have a simple expression for the same series 
starting at k=b, that is:

         infinity  
           sum    a^k/k! = ?
           k=b

Thank you for information,

Jean-Francois Hellings
Project Engineer, SAIT Systems
Brussels, Belgium

Date: 8/3/96 at 9:33:47
From: Doctor Jerry
Subject: Re: Math Problem

Are you asking if there is a simple expression for the tail of the 
exponential series - if there is a closed form for the series

a^b/b!+a^(b+1)/(b+1)!+ . . .      ?

Since

e^a = 1+a/1!+a^2/2!+...+a^(b-1)/(b-1)!+a^b/b!+a^(b+1)/(b+1)! + ...,

we can write

a^b/b!+a^(b+1)/(b+1)! + ... = 
   e^a - (1+a/1!+a^2/2!+...+a^(b-1)/(b-1)!).

This is a closed form for the series on the left.  If b were large, 
this might not be much help.  


The question as to whether there is a special function giving the 
value of

a^b/b!+a^(b+1)/(b+1)! + ... 

remains.  We worked on this question for a while, trying various 
things and looking in a book of special functions. We found nothing, 
but an expert in special functions might be able to answer your 
question.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/