Identify function based on known Taylor series

Question

I've come across the following Taylor series, to which I hope there exist an analytic solution:

$$f(x) = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k}}{(2k+c)\cdot(2k)!}$$

The parameter $c$ is some strictly positive real number. The only thing I managed to find out (based partly on plotting the Taylor series numerically) is that $c=1 \implies f(x) = \frac{\rm{sin(x)}}{x} $, and that it appears to be that $\lim\limits_{c\to \infty}(f(x)) = c\cdot\rm{sin(x)}$

Is there a general function having this Taylor series for any (positive) value of $c$?

Answer 1

This is in fact $$f_c(x) = \sum_{k=0}^{\infty} \frac{(-1)^k x^{2k}}{(2k+c)\cdot(2k)!}=\frac{1}{c}\, _1F_2\left(\frac{c}{2};\frac{1}{2},\frac{c}{2}+1;-\frac{x^2}{4}\right)$$ where appears the generalized hypergeometric function.

For some integer values of $c$, here are the expressions $$f_1(x)=\frac{\sin (x)}{x}$$ $$f_2(x)=\frac{x \sin (x)+\cos (x)-1}{x^2}$$ $$f_3(x)=\frac{x^2 \sin (x)+2 x \cos (x)-2 \sin (x)}{x^3}$$ $$f_4(x)=\frac{x^3 \sin (x)+3 x^2 \cos (x)-6 x \sin (x)-6 \cos (x)+6}{x^4}$$ $$f_5(x)=\frac{x^4 \sin (x)+4 x^3 \cos (x)-12 x^2 \sin (x)-24 x \cos (x)+24 \sin (x)}{x^5}$$ where you an see some patterns.

Answer 2

To complete Claude Leibovici's insightful answer, you may notice that $f_c(x)$ is an entire function with the following integral representation:

$$ x^c\,f_c(x) = \int_{0}^{x}\sum_{k\geq 0}\frac{(-1)^k}{(2k)!}z^{2k+c-1}\,dz = \int_{0}^{x} z^{c-1}\cos(z)\,dz \tag{1}$$ so: $$ f_c(x) = x \int_{0}^{1} u^{c-1}\cos(xu)\,du \tag{2}$$ and in terms of the Laplace transform: $$ \left(\mathcal{L} f_c\right)(s) = \Gamma(c)\,\frac{(s+i)^c+(s-i)^c}{2s(1+s^2)^c}=\frac{\Gamma(c)}{2s}\left(\frac{1}{(s+i)^c}+\frac{1}{(s-i)^c}\right)\tag{3}$$ so $f_c(x)$ is the inverse Laplace transform of a simple function.