Re: Is there any quick way to calculate this series?

On Mar 13, 2:50 am, Chris <ch...@xxxxxxxxxxxxx> wrote:
Apologies,

I wrote the series down incorrectly, it is as follows:

-((y^x)/x) - ((y(^x+1)/(x+1)) - ((y^(x+2)/(x+2)) ... - ((y^(x+n)/(x-
n))

Thanks,

On Mar 13, 9:37 am, Chris <ch...@xxxxxxxxxxxxx> wrote:

Dear sci.math,

I have the following series:

-((y^x)/x) - ((y(^x-1)/(x-1)) - ((y^(x-2)/(x-2)) ... - ((y^(x-n)/(x-
n))

Is there an alternative single formula I can use to replace the above?

Thank you very much for any help.

Regards,

Chris.



Maple 9.5 gets:

S:=sum(y^(k+i)/(k+i),i=0..n);
k (k + n)
S := y LerchPhi(y, 1, k) - LerchPhi(y, 1, k + n) y

(k + n)
y
+ --------
k + n

t:=LerchPhi(y,1,k);

t := LerchPhi(y, 1, k)

convert(t,'hypergeom');

hypergeom([1, k], [k + 1], y)
-----------------------------
k

s:=LerchPhi(y,1,k+n);

s := LerchPhi(y, 1, k + n)

convert(s,'hypergeom');

hypergeom([1, k + n], [k + n + 1], y)
-------------------------------------
k + n

Do, you can either express the answer in terms of the LerchPhi
function, or else in terms of hypergeometric functions.


Here is a definition from the Maple help pages:
The Lerch Phi function is defined as follows:
infinity
------ n
\ z
LerchPhi(z,a,v) = > -------
/ a
------ (v+n)
n = 0
This definition is valid for abs(z) < 1 or abs(z) = 1 and Re(a) > 1.
By analytic continuation, it is extended to the whole complex z-plane
for each value of a and v.


R.G. Vickson
.

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