I have strong feeling that the above function $$f_\alpha (x) = \sum_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)}$$ is a known special function but I can't seem to recognize it. Here $\Gamma(x)$ denotes the extension of the factorial. I want to know if anyone can recognize this.

I have strong feeling that the function, $$ f_\alpha (x) = \sum_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)}, $$ is a known special function (here $\Gamma(x)$ is the usual extension of the factorial). Is this the case?

I have strong feeling that the above function $$f_\alpha (x) = \sum_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)}$$ is a known special function but I can't seem to recognize it. Here $\Gamma(x)$ denotes the extension of the factorial. I want to know if anyone can recognize this.

Thanks

I have strong feeling that the above function $$f_\alpha (x) = \sum_{n=0}^\infty \frac{x^n}{n!\Gamma(1+n\alpha)}$$ is a known special function but I can't seem to recognize it. Here $\Gamma(x)$ denotes the extension of the factorial. I want to know if anyone can recognize this.