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

In Mathworld's article Gamma function, in line (96), we find the formula,

$\sum_{k=0}^\infty (8k+1)\left(\frac{\Gamma(k+\frac{1}{4})}{k!\;\Gamma(\frac{1}{4})}\right)^4 = 2^{3/2}\frac{1}{\sqrt{\pi}\,\left(\Gamma(3/4)\right)^2}$

On a whim, I evaluated the LHS and RHS using Mathematica to 100-digit precision, and found the first few digits as,

$\text{LHS} = 1.062679901\dots$

$\text{RHS} = 1.062679899\dots$

Ahem, they don't match. If it is a typo, then I find it interesting it is exceedingly close.

So what is the problem? 1) Did I input it in Mathematica wrongly? 2) Is there a typo, or misplaced symbol by authors after Ramanujan (Weisstein gives Hardy et al as references) 3) Or was Ramanujan just mistaken?

share|cite|improve this question
up vote 7 down vote accepted

The series in question converges slowly, $(8k+1) \left( \frac{(1/4)_k}{k!} \right)^4 \sim \frac{8}{k^2 \Gamma^4(1/4)}$, hence it may be that you have not computed enough terms.

The sum represents a value of a hypergeometric function: $$ \sum_{k=0}^\infty (8k+1) \left( \frac{(1/4)_k}{k!} \right)^4 = {}_4F_3\left( \frac{1}{4},\frac{1}{4}, \frac{1}{4}, \frac{1}{4}; 1,1,1 | 1\right) - \frac{1}{32} {}_4F_3\left( \frac{5}{4},\frac{5}{4}, \frac{5}{4}, \frac{5}{4}; 2,2,2 | 1\right) $$ Evaluating these numerically agrees with the expression in terms of $\Gamma$ constant:

In[18]:= Sum[(8 k + 1) (Pochhammer[1/4, k]/k!)^4, {k, 0, \[Infinity]}]

Out[18]= 1/32 (32 HypergeometricPFQ[{1/4, 1/4, 1/4, 1/4}, {1, 1, 1}, 
     1] + HypergeometricPFQ[{5/4, 5/4, 5/4, 5/4}, {2, 2, 2}, 1])

In[19]:= N[%, 30]

Out[19]= 1.06267989991684365118249019510

In[20]:= N[(2^(3/2)/(Sqrt[Pi] Gamma[3/4]^2)), 30]

Out[20]= 1.06267989991684365118249019510
share|cite|improve this answer
Thanks, Sasha. But this is strange. I typed 1/32 (32 HypergeometricPFQ[{1/4, 1/4, 1/4, 1/4}, {1, 1, 1}, 1] + HypergeometricPFQ[{5/4, 5/4, 5/4, 5/4}, {2, 2, 2}, 1]) into my old Mathematica (I'm still using version 4) with N[%,100], and I still get that erroneous value. :-( – Tito Piezas III May 7 '12 at 20:45
@TitoPiezasIII What version of Mathematica did you use? – Sasha May 7 '12 at 20:46
An old one. Version 4. There's a bug, right? – Tito Piezas III May 7 '12 at 20:48
Numerical evaluation of ${}_{p+1}F_{p}(z)$ for $z$ near the unit circle has much improved since version 4. – Sasha May 7 '12 at 20:52
Good God, and here I thought it was infallible. I better buy a new one then. (If I can scrape up the dough.) :-( – Tito Piezas III May 7 '12 at 20:56

FWIW, Maple 16 gets the sum right (symbolically, and numerically to within a reasonable roundoff error).

L:= Sum((8*k+1)*(GAMMA(k+1/4)/k!/GAMMA(1/4))^4,k=0..infinity);

R:= value(L);

$$ R := 2\,{\frac {\sqrt {2}}{\sqrt {\pi } \left( \Gamma \left( 3/4 \right) \right) ^{2}}}$$

evalf(L - R, 100);

$$ 0.1\ 10^{-98} $$

share|cite|improve this answer
Thanks, Dr. Israel. I was using an old Mathematica (Version 4) and unfortunately, even with 100-digit precision, it gave me an erroneous value. Close, but not quite. – Tito Piezas III May 7 '12 at 20:59

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.