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A generating function is a way of encoding an infinite sequence of numbers by treating them as the coefficients of a formal power series. Tag questions involving generating functions in this.

2 votes
0 answers
154 views

Unrestricting The Parameters of a Functional Equation

Good evening. I am looking into methods of generalization of Bernoulli polynomials. First, define $$\Phi_{N,k}(x)=\frac{1}{N}\sum_{j=0}^{N-1}\omega_N^{-jk}\exp\left(\omega_N^jx\right)$$ where $\o …
3 votes
0 answers
272 views

Combination of Generating Functions

Suppose I have the following generating functions: $$\frac{x^ke^{\left(z-\frac{1}{N}\right)x}}{N^{k-1}k!\sum_{j=0}^{N-1}w_N^{-jk}e^{\frac{w_N^jx}{N}}}=\sum_{j=0}^\infty H_{N,k,j}(z)\frac{x^j}{j!}$$ w …
2 votes
1 answer
213 views

Coefficients for Powers of the Mittag-Leffler Function

Considering the one parameter Mittag-Leffler function, $$E_{\alpha}(z)=\sum_{k=0}^\infty\frac{z^{k}}{\Gamma(\alpha k+1)}, \Re(\alpha)>0$$ Considering then the generating function for $E_\alpha(z^\a …