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Results tagged with generating-functions
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user 60457
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
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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
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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
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answer
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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 …