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TableĀ 7 Results for \({}_{e^{(s)}}\mathit{F}^{(s,\upsilon )}_{n}(x,y,z)\)

From: Construction of a new family of Fubini-type polynomials and its applications

Generating function \(\frac{2^{\upsilon }}{(2-e^{t})^{2\upsilon }} (\frac{e^{xt}}{1-yt^{s}} ) \sin (zt)= \sum_{n=0}^{\infty } {}_{e^{(s)}}\mathit{F}^{(s,\upsilon )}_{n}(x,y,z) \frac{ t^{n}}{n!} \)
Multiplicative and derivative operators \(\hat{M}_{TE\mathit{F}s}=x+\frac{syD_{x}^{s-1}}{1-yD_{x}^{s}}+\frac{2\upsilon e^{D_{x}}}{2-e^{D_{x}}}+z \cot (zD_{x})\), \(\hat{P}_{TE\mathit{F}s}:= D_{x}\)
Differential equation \((xD_{x}+\frac{syD_{x}^{s}}{1-yD_{x}^{s}}+\frac{2\upsilon e^{D_{x}}}{2-e^{D_{x}}}D_{x}+z \cot (zD_{x})D_{x}-n ){}_{e^{(s)}}\mathit{F}^{(s,\upsilon )}_{n}(x,y,z)=0\)
Identities and relations \({}_{e^{(s)}}\mathit{F}^{(s,\upsilon )}_{n}(x,y,z)=\sum_{\kappa =0}^{n} \binom{n }{\kappa } \mathit{F}^{(s,\upsilon )}_{n-\kappa }(z) e^{(s)}_{\kappa }(x,y)\)
\({}_{e^{(s)}}\mathit{F}^{(s,\upsilon )}_{n}(x+u,y,z)=\sum_{\kappa =0}^{n} \binom{n }{\kappa } \mathit{F}^{(s,\upsilon )}_{n-\kappa }(x,z) e^{(s)}_{\kappa }(u,y)\)
\({}_{e^{(s)}}\mathit{F}^{(s,\upsilon +\sigma )}_{n}(x,y,z)=\sum_{\kappa =0}^{n} \binom{n }{\kappa } \mathit{F}^{(\upsilon )}_{\kappa } {}_{e^{(s)}}\mathit{F}^{(s,\sigma )}_{n-\kappa }(x,y,z)\)
\((\frac{x^{n}}{1-yt^{s}} ) \sin (zt)=\sum_{\delta =0}^{2\upsilon }\sum_{\kappa =0}^{n}(-1)^{\delta } \binom{2\upsilon }{\delta }\binom{n }{\kappa }2^{\upsilon -\delta } \delta ^{n-\kappa } {}_{e^{(s)}}\mathit{F}^{(s,\upsilon )}_{\kappa }(x,y,z)\)
Partial derivatives equations \(\frac{\partial ^{m}}{\partial x^{m}} \lbrace {}_{e^{(s)}}\mathit{F}^{(s,\upsilon )}_{n}(x,y,z) \rbrace =2^{-\frac{\delta }{2}} \sum_{\kappa =0}^{n}\frac{m! \kappa !}{(\kappa +\delta )!} \binom{n }{\kappa }\binom{n-\kappa }{m} \mathfrak{B}^{(\delta )}_{\kappa +\delta } (\frac{1}{2} ) {}_{e^{(s)}}\mathit{F}^{ (s,\upsilon -\frac{\delta }{2} )}_{n-\kappa -m}(x,y,z)\)
\(\frac{\partial ^{m}}{\partial u^{m}} \lbrace {}_{e^{(s)}}\mathit{F}^{(s,\upsilon +\sigma )}_{n}(x+u,y,z) \rbrace = \frac{n!}{2^{3\upsilon } (n+2\upsilon )!} \sum_{\kappa =0}^{n+2\upsilon }m! \binom{n+2\upsilon }{\kappa }\binom{\kappa }{m} \mathcal{G}^{(2\upsilon )}_{\kappa -m} (x,-\frac{1}{2} ) {}_{e^{(s)}}\mathit{F}^{(s,\sigma )}_{n+2\upsilon -\kappa }(u,y,z)\)
\(\frac{\partial }{\partial x} \lbrace {}_{e^{(s)}}\mathit{F}^{(s,\upsilon )}_{n}(x,y,z) \rbrace =\frac{1}{2^{3\delta }} \sum_{\kappa =0}^{n}\frac{(n-\kappa ) \kappa !}{(\kappa +2\delta )!}\binom{n }{\kappa } \mathcal{G}^{(2\delta )}_{\kappa +2\delta } (-\frac{1}{2} ) {}_{e^{(s)}}\mathit{F}^{(s,\upsilon -\delta )}_{n-\kappa -1}(x,y,z)\)