I would like to have the following sequence, $(a)_n$, in a series where $a_0 = 1$ and $a_n = a(a+1)\cdots (a+ n -1)$. The series I have is $$ 2\sum_{n=0}^{\infty}\frac{(1/2)_n}{n!(2n+1)^3} $$ I know how to represent series in mathematica but I have no idea how I can insert the sequence into the series code. What I have is

2 Sum[Sequence Here/(n!(2n+1)^3),{n,0,infinity}]

Is it possible to do this in Mathematica?

  • $\begingroup$ Are you looking for Pochhammer perhaps? $\endgroup$ – ciao Feb 21 '15 at 1:52
  • $\begingroup$ @rasher yes, that worked. Should I delete or do you want provide an answer? $\endgroup$ – dustin Feb 21 '15 at 1:54
  • $\begingroup$ Sure, I'll answer... future reference. $\endgroup$ – ciao Feb 21 '15 at 1:56

You're after Pochhammer:

Pochhammer[a, 5]

(* a (1 + a) (2 + a) (3 + a) (4 + a) *)

Do note, the Mathematica definition by default is opposite that used in many fields: it is the Rising Factorial, while the traditional form can be confused with Falling Factorial, the perhaps more common use of Pochhammer...

  • $\begingroup$ @2012rcampion Oops... $\endgroup$ – ciao Feb 21 '15 at 2:38

You can also construct that series yourself, should you need that in another context in the future:


and insert that into your formula:

2 Sum[sa[n]/(n!(2 n+1)^3), {n,0,\[Infinity]}]

or avoid the function and write:

2 Sum[Product[a+i,{i,0,n-1}]/(n!(2 n+1)^3), {n,0,\[Infinity]}]

both giving you the result:

(1/(16 Gamma[3/2-a]))Sqrt[\[Pi]]
  Gamma[1-a] (\[Pi]^2+2 PolyGamma[0,1/2]^2 - 
   4 PolyGamma[0,1/2] PolyGamma[0,3/2-a] + 
   2 PolyGamma[0,3/2-a]^2 - 2 PolyGamma[1,3/2-a])

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