Asymptotic formula for a power series

[zetafunction]

where can i find a proof of the following identity ?

$$\sum_{n=0}^{\infty} (-x)^{n} \frac{c(n)}{n!} \sim c(x) +(1/2)c''(x)x+(1/6)c'''(x)x + (1/8)x^{2}c'''' (x) +++++$$

 

[ross_tang]

Can you give a full form of right-hand side? And are you sure your identity must be right? I get different result from yours.

$$\sum _{n=0}^{\infty } \frac{(-x)^nc(n)}{n!}=\sum _{k=0}^{\infty } \left(\sum _{n=0}^{\infty } \frac{(-x)^n(n-x)^k}{n!k!}\right)c^{(k)}(x)$$

For the first few terms:
$$e^{-x} c(x)-2 e^{-x} x c^{(1)}(x)+\frac{1}{2} e^{-x} x (-1+4 x) c^{(2)}(x)-\frac{1}{6} e^{-x} x \left(1-6 x+8 x^2\right) c^{(3)}(x)+\frac{1}{24} e^{-x} x \left(-1+11 x-24 x^2+16 x^3\right) c^{(4)}(x)+\text{...}$$

Even if I use x is large, I can't get your result.