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TableĀ 5 Results for \({}_{\mathtt{L}}\mathit{F}^{(s,\upsilon )}_{n}(y,x,z)\)

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

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