4
$\begingroup$

$\prod _{n=1}^{\infty } \left(1+a (c+n)^b\right)$ where a > 0, b < -1, and c >= 0

Is there a special function, series expansion, or other simpler (or maybe just interesting) representation of this infinite product ?

$\endgroup$

1 Answer 1

2
$\begingroup$

This is but a partial answer.

If $c$ is integer, then you can add a few terms to reduce the problem to $\prod_{n=1}^\infty (1+an^b)$. If in addition $b$ is integer, then you can write $$(1+an^b) = \prod_{\xi^b=-a}(1+(-\frac \xi n))$$ and reduce to a finite product of inverses of $\Gamma$ functions and exponentials, using the product formula for $\Gamma$ (at complex values of the parameter).

I don't know what to make of the general case, but there are more knowledgeable people here.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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