Table 4 Results for \({}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{n}(y,x,z)\)

Generating function

\(\frac{2^{\upsilon }}{(2-e^{t})^{2\upsilon }} e^{xt} C_{0}(yt) \cos (zt)= \sum_{n=0}^{\infty } {}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{n}(y,x,z) \frac{ t^{n}}{n!} \)

Multiplicative and derivative operators

\(\hat{M}_{L\mathit{F}c}=x-D^{-1}_{y}+\frac{2\upsilon e^{D_{x}}}{2-e^{D_{x}}}-z \tan (zD_{x})\), \(\hat{P}_{L\mathit{F}c}:= D_{x}\)

Differential equation

\((xD_{x}-D^{-1}_{y}D_{x}+\frac{2\upsilon e^{D_{x}}}{2-e^{D_{x}}}D_{x}-z \tan (zD_{x})D_{x}-n ){}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{n}(y,x,z)=0\)

Identities and relations

\({}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{n}(y,x,z)=\sum_{\kappa =0}^{n} \binom{n }{\kappa } \mathit{F}^{(c,\upsilon )}_{n-\kappa }(z) \mathtt{L}_{\kappa }(y,x)\)

\({}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{n}(y,x+u,z)=\sum_{\kappa =0}^{n} \binom{n }{\kappa } {}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{n-\kappa }(y,x,z) u^{\kappa }\)

\({}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{n+\kappa }(\omega ,x,z)=\sum_{l=0}^{n}\sum_{m=0}^{\kappa } \binom{n }{l} \binom{\kappa }{m} (\omega -x)^{l+m} {}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{n+\kappa -l-m}(y,x,z)\)

\({}_{\mathtt{L}}\mathit{F}^{(c,\upsilon +\sigma )}_{n}(y,x,z)=\sum_{\kappa =0}^{n} \binom{n }{\kappa } \mathit{F}^{(\upsilon )}_{\kappa } {}_{\mathtt{L}}\mathit{F}^{(c,\sigma )}_{n-\kappa }(y,x,z)\)

\(x^{n} C_{0}(yt) \cos (zt)=\sum_{\delta =0}^{2\upsilon }\sum_{\kappa =0}^{n}(-1)^{\delta } \binom{2\upsilon }{\delta }\binom{n }{\kappa }2^{\upsilon -\delta } \delta ^{n-\kappa } {}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{\kappa }(y,x,z)\)

Partial derivatives equations

\(\frac{\partial ^{m}}{\partial x^{m}} \lbrace {}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{n}(y,x,z) \rbrace =2^{-\frac{3\delta }{2}} \sum_{\kappa =0}^{n}m! \binom{n }{\kappa }\binom{n-\kappa }{m} \mathcal{E}^{(\delta )}_{\kappa } (-\frac{1}{2} ) {}_{\mathtt{L}}\mathit{F}^{ (c,\upsilon -\frac{\delta }{2} )}_{n-\kappa -m}(y,x,z)\)

\(\frac{\partial ^{m}}{\partial u^{m}} \lbrace {}_{\mathtt{L}}\mathit{F}^{(c,\upsilon +\sigma )}_{n}(y,x+u,z) \rbrace = \frac{n!}{2^{\upsilon } (n+2\upsilon )!} \sum_{\kappa =0}^{n+2\upsilon }m! \binom{n+2\upsilon }{\kappa }\binom{\kappa }{m} \mathfrak{B}^{(2\upsilon )}_{\kappa -m} (x,\frac{1}{2} ) {}_{\mathtt{L}}\mathit{F}^{(c,\sigma )}_{n+2\upsilon -\kappa }(u,x,z)\)

\(\frac{\partial }{\partial x} \lbrace {}_{\mathtt{L}}\mathit{F}^{(c,\upsilon )}_{n}(y,x,z) \rbrace =\frac{1}{2^{3\delta }} \sum_{\kappa =0}^{n}(n-\kappa )\binom{n }{\kappa } \mathcal{E}^{(2\delta )}_{\kappa } (-\frac{1}{2} ) {}_{\mathtt{L}}\mathit{F}^{(c,\upsilon -\delta )}_{n-\kappa -1}(y,x,z)\)