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

$$\begin{align*} \Sigma_{k=0}^{\infty}\frac{a^k}{(k-x)!} & = a^x \Sigma_{k=0}^{\infty}\frac{a^{k-x}}{(k-x)!} \\ & = a^x \Sigma_{k=-x}^{\infty} \frac{a^k}{k!} \\ & = a^x \left( e^a - \Sigma_{k = 0}^{-x-1} \frac{a^k}{k!} \right) \end{align*}$$​

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