Series Dependencies

Question

If i have a series $$ a_n = \sum_{n=0}^\infty \frac{x^{2n}}{b^nn!}(n+c)^2 $$

and another series $$ c_n = \sum_{n=0}^\infty \frac{x^{2n}}{n!}(n+c)^2 $$

Is there some way to find some $f(x)$ so that

$$ a_n = f(x,b)c_n $$

Thank you

Answer 1

First note that what you called $a_n$ actually does not depend on $n$, but on $x$, $b$ and $c$. The same applied to $c_n$. Now, we have: $$ a(x,b,c):= \sum_{n=0}^{\infty} \frac{x^{ 2n} }{b^n n!} (n+c)^2 $$ and $$ r(x,c):= \sum_{n=0}^{\infty} \frac{x^{ 2n} }{ n!} (n+c)^2 $$ Then, it is clear that: $$ a(x,b,c) = r \left(\frac{ x}{ \sqrt{ b}}, c \right ) $$ where we assume $b >0$.