Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

I am hoping to evaluate a simpler expression for the following: $${}_3F_2(-n,1,1; a, (a+3)/2; -1)$$ Here $n,a \in \mathbb{N}$ and $a$ is odd. I am also interested in the asymptotics in $n \in \mathbb{N}$ for fixed $a$. There are numerous transformations for hypergeometric series with unit argument, but I was not able to find any with negative unity as the argument. Considering that two of the parameters in the numerator are unity, I was hoping that a nice expression would be nearby.

I would appreciate if someone can help with references or ideas.

share|cite|improve this question
up vote 2 down vote accepted

With $f_n$ denoting the hypergeometric function, it can be checked to satisfy the inhomogeneous rank 2 recurrence equation: $$ (n+a+1)\left(n+\frac{a+5}{2}\right)f_{n+2} - 3 (n+2)\left(n+\frac{a+3}{2}\right) f_{n+1} + 2 (n+1)(n+2) f_n = \frac{a^2-1}{2} $$

In[52]:= Table[-(1/2) (a^2 - 1) + (n + a + 1) (n + (a + 3)/2 + 1) y[
      n + 2] - 3 (n + 2) (n + (a + 3)/2) y[n + 1] + 
    2 (n + 1) (n + 2) y[n] /. 
   y -> Function[k, 
     HypergeometricPFQ[{-k, 1, 1}, {a, (a + 3)/2}, -1]], {n, 0, 
   6}] // Together

Out[52]= {0, 0, 0, 0, 0, 0, 0}

From this recurrence equation, using techniques of Birkhoff and Trjitzinsky, "Analytic theory of singular difference equations", Acta Math., 60 (1932), pp. 1–89, one gets: $$ f_n = 2^n n^{(1-3a)/2} \left(c_1 + \mathcal{O}\left(n^{-1}\right) \right) + \frac{1}{n} \left(c_2 + \mathcal{O}\left(n^{-1}\right) \right) + \frac{a^2-1}{2} \frac{\log(n)}{n} \left(1 + \mathcal{O}\left(n^{-1}\right) \right) $$

Also note, that for special case of $a=1$, closed-form is easy to find: $$ f_n = {}_3F_2\left(-n,1,1; 1,2; -1\right) = {}_2F_1\left(-n,1; 2; -1\right) = \frac{2^{n+1}-1}{n+1} $$

One could also try use the following integral representation for $f_n$, valid for $a>1$: $$ f_n = \frac{a+1}{2} \int_0^1 \left(1-t\right)^{(a-1)/2} \cdot {}_2F_1\left(-n, 1; a; -t\right) \mathrm{d} t $$

In[79]:= Table[
 Simplify[HypergeometricPFQ[{-n, 1, 1}, {a, (a + 3)/2}, -1] == 
    a > 1, ((a + 1)/2)*
     Integrate[(1 - t)^((a - 1)/2)*
       HypergeometricPFQ[{-n, 1}, {a}, -t], {t, 0, 1}]]], {n, 1, 5}]

Out[79]= {True, True, True, True, True}
share|cite|improve this answer
Wow! Thanks @Sasha! I did not know any such techniques were available. Could you suggest a book for the techniques you have employed from Birkoff and Trjitzinsky? – Ankur Sep 19 '12 at 5:14
@Ankur I do not know of such book, which is not to say one does not exist. You could also check Birkoff's article, freely available from JSTOR. – Sasha Sep 19 '12 at 15:24
Could you tell which theorem from Birkoff and Trjitzinsky have you employed here? I went through the paper but unable to find a result that would suit my problem. Also, I think we are in the same town - if that's ok I could you even meet you in person to get your help. – Ankur Sep 21 '12 at 4:18
@Ankur Well, I might have given you not quite the right article reference. See Bikhoff's article, "Formal theory of irregular singular difference equations", Acta Mathematica ( SpringerLink). We are indeed in the same town. Small world. – Sasha Sep 21 '12 at 5:06
@Ankur Since the inhomogeneous recurrence equation reads $g_n = \frac{a^2-1}{2}$, it can be homogenized, as $g_{n+1}-g_n = 0$. This implies order 3 recurrence homogeneous recurrence equation for $f_n$. You can apply similar technique to find the asymptotic behavior of $f_n$. Substitution back into the inhomogeneous system determines relation between two of three free coefficients. I hope this is clear enough. – Sasha Oct 2 '12 at 1:02

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.