- #1
sarrah1
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This was posted to calculus forum. I suppose it should have been posted here.
I am trying to find a closed form expression/or limit as $n\implies\infty$ of
${S}_{n}=\sum_{k=0}^{n}{n \choose k} {a}^{k} \sum_{j=0}^{n}{n \choose j}\frac{{b}^{n-j}{c}^{j}}{(k+j)!}$
where $a$ , $b$ and $c$ are positive constants
the ultimate aim is to find the limit of ${S}_{n}$ as $n$ tends to infinity and that this limit is zero under some conditions imposed upon the constants a,b,c.
Euge gave me once the idea for a similar expression by taking $k+j$ outside the 2nd sum and the remaining becomes equal to ${(b+c)}^{n}$ . As for the first term, when divided by $k!$ it will be less or equal to
$\sum_{k=0}^{n}{n \choose k} {a}^{k}/k!$
So if this one converges to some sum that tends to zero, OR Instead to a finite value, But $b+c<1$ then mission done and ${S}_{n}$ tends to zero.
On a second thought the expression $\sum_{k=0}^{n}{n \choose k} {a}^{k}/k!$ is a Laguerre polynomial ${L}_{n}(-a) , a>0 $ which diverges $n\implies\infty$ . so one has to rely on the whole expression ${S}_{n}$ above. If someone knows if there is some double Laguerre, or product of 2 laguerre polynomials or double binomial transform, etc...
I shall be grateful
happy new prosperous year
special regards to Euge, Oplag and Akbach
sarrah
I am trying to find a closed form expression/or limit as $n\implies\infty$ of
${S}_{n}=\sum_{k=0}^{n}{n \choose k} {a}^{k} \sum_{j=0}^{n}{n \choose j}\frac{{b}^{n-j}{c}^{j}}{(k+j)!}$
where $a$ , $b$ and $c$ are positive constants
the ultimate aim is to find the limit of ${S}_{n}$ as $n$ tends to infinity and that this limit is zero under some conditions imposed upon the constants a,b,c.
Euge gave me once the idea for a similar expression by taking $k+j$ outside the 2nd sum and the remaining becomes equal to ${(b+c)}^{n}$ . As for the first term, when divided by $k!$ it will be less or equal to
$\sum_{k=0}^{n}{n \choose k} {a}^{k}/k!$
So if this one converges to some sum that tends to zero, OR Instead to a finite value, But $b+c<1$ then mission done and ${S}_{n}$ tends to zero.
On a second thought the expression $\sum_{k=0}^{n}{n \choose k} {a}^{k}/k!$ is a Laguerre polynomial ${L}_{n}(-a) , a>0 $ which diverges $n\implies\infty$ . so one has to rely on the whole expression ${S}_{n}$ above. If someone knows if there is some double Laguerre, or product of 2 laguerre polynomials or double binomial transform, etc...
I shall be grateful
happy new prosperous year
special regards to Euge, Oplag and Akbach
sarrah