Consider the given sum.
$$\sum^{\infty}_{n=0} \frac{x^n}{[(2n+1)!]^3} = ?$$ Does there exist a closed form of the above summation. What is the general procedure to perform such a sum ? I tried writing the above in terms of hypergeometrics but wasn't very successful. Any ideas
Consider the ratio of consecutive terms:
$$\frac{a_{n+1}}{a_n}=\frac{x}{(2n+2)^3(2n+3)^3}=\frac{x/64}{(n+1)^3(n+\frac32)^3}$$
with $a_0=1$, and hence, the series is given by the generalized hypergeometric function:
$$\sum_{n=0}^\infty\frac{x^n}{[(2n+1)!]^3}={}_0F_5\left(;1,1,\frac32,\frac32,\frac32;\frac x{64}\right)$$
I doubt there is anything one can do here, save for possibly some special values (that I wouldn't know of). In general, converting a series such as this to a generalized hypergeometric function simply amounts to computing the ratio of consecutive terms and factoring the numerator and denominator, and extra constants go into the last argument.