Does anyone recognise this serie?

[_joey]

\Sigma_{k=0}^{\infty}\frac{a^k}{(k-x)!}

Thanks!

 

[sylas]

\Sigma_{k=0}^{\infty}\frac{a^k}{(k-x)!}

Thanks!

If x is positive, then you have factorials of a negative number, which is a tad unusual.

If x is not an integer, then you have factorials of a non-integer. Also unusual.

If x is a negative integer, you have

<br /> \begin{align*}<br /> \Sigma_{k=0}^{\infty}\frac{a^k}{(k-x)!} &amp; = a^x \Sigma_{k=0}^{\infty}\frac{a^{k-x}}{(k-x)!} \\<br /> &amp; = a^x \Sigma_{k=-x}^{\infty} \frac{a^k}{k!} \\<br /> &amp; = a^x \left( e^a - \Sigma_{k = 0}^{-x-1} \frac{a^k}{k!} \right)<br /> \end{align*}​

This is the exponential function, scaled and translated.

P.S. Added in edit. Bad description there sorry. It is not scaled and translated by a constant. You subtract a polynomial, and then divide by a^-x.

Cheers -- sylas