A new class of Gould-Hopper-Eulerian-type polynomials

Figures & data

Table 1. Results of the parametric kind Gould-Hopper-Apostol-Euler polynomials.

Generating functions	${\left(\frac{2}{\lambda {\mathrm{e}}^{\tau }+1}\right)}^{\alpha }{\mathrm{e}}^{u\tau +v{\tau }^{\sigma }}\cos \left(w\tau \right)=\sum _{\kappa =0}^{\mathrm{\infty }}{}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda )\frac{{\tau }^{\kappa }}{\kappa !},$
	${\left(\frac{2}{\lambda {\mathrm{e}}^{\tau }+1}\right)}^{\alpha }{\mathrm{e}}^{u\tau +v{\tau }^{\sigma }}\sin \left(w\tau \right)=\sum _{\kappa =0}^{\mathrm{\infty }}{}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda )\frac{{\tau }^{\kappa }}{\kappa !}.$
Series	${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda )=\sum _{r=0}^{\kappa }(\genfrac{}{}{0ex}{}{\kappa }{r}){\mathcal{E}}_r^{\left(\alpha \right)}\left(\lambda \right){\mathit{G}}_{\kappa -r}^{(c,\sigma )}(u,v,w)$,
representations	${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda )=\sum _{r=0}^{\kappa }(\genfrac{}{}{0ex}{}{\kappa }{r}){\mathcal{E}}_r^{\left(\alpha \right)}\left(\lambda \right){\mathit{G}}_{\kappa -r}^{(s,\sigma )}(u,v,w)$.
Multiplicative and	${\hat{M}}_{{}_{{\mathit{G}}^{\left(\sigma \right)}}\mathcal{E}c}=u+\sigma v{D}_u^{\sigma -1}-\frac{\alpha \lambda {\mathrm{e}}^{D_u}}{\lambda {\mathrm{e}}^{D_u}+1}-w\tan \left(w{D}_u\right);{\hat{P}}_{{}_{{\mathit{G}}^{\left(\sigma \right)}}\mathcal{E}c}=D_u,$
derivative operators	${\hat{M}}_{{}_{{\mathit{G}}^{\left(\sigma \right)}}\mathcal{E}s}=u+\sigma v{D}_u^{\sigma -1}-\frac{\alpha \lambda {\mathrm{e}}^{D_u}}{\lambda {\mathrm{e}}^{D_u}+1}+w\cot \left(w{D}_u\right);{\hat{P}}_{{}_{{\mathit{G}}^{\left(\sigma \right)}}\mathcal{E}s}=D_u.$
Differential equations	$(u{D}_u+\sigma v{D}_u^{\sigma }-\frac{\alpha \lambda {\mathrm{e}}^{D_u}}{\lambda {\mathrm{e}}^{D_u}+1}{D}_u-w\tan (w{D}_u)D_u-\kappa ){}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda )=0,$
	$(u{D}_u+\sigma v{D}_u^{\sigma }-\frac{\alpha \lambda {\mathrm{e}}^{D_u}}{\lambda {\mathrm{e}}^{D_u}+1}{D}_u+w\cot (w{D}_u)D_u-\kappa ){}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda )=0$.
Some useful	${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda )=\kappa !\sum _{r=0}^{\left[\frac{\kappa }{\sigma }\right]}\frac{{\mathcal{E}}_{\kappa -\sigma r}^{(c,\alpha )}(u,w;\lambda )v^r}{(\kappa -\sigma r)!r!}$,
Formulae	${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda )=\kappa !\sum _{r=0}^{\left[\frac{\kappa }{\sigma }\right]}\frac{{\mathcal{E}}_{\kappa -\sigma r}^{(s,\alpha )}(u,w;\lambda )v^r}{(\kappa -\sigma r)!r!}$,
	${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(s,\alpha )}(u+z,v,w;\lambda )=\sum _{r=0}^{\kappa }(\genfrac{}{}{0ex}{}{\kappa }{r}){\mathcal{E}}_{\kappa -r}^{(s,\alpha )}(z,w;\lambda ){\mathit{G}}_r^{\left(\sigma \right)}(u,v)$,
	${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(s,\alpha +\text{β})}(u+x,v,w+y;\lambda )$
	$=\sum _{r=0}^{\kappa }(\genfrac{}{}{0ex}{}{\kappa }{r})\left[{}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa -r}^{(s,\alpha )}\right(u,v,w;\lambda \left){\mathcal{E}}_r^{(c,\text{β})}\right(x,y;\lambda ,\nu )+{\mathcal{E}}_{\kappa -r}^{(s,\alpha )}(u,y;\lambda \left){}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_r^{(c,\text{β})}\right(x,v,w;\lambda \left)\right]$,
	${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{\left(\alpha \right)}(u+iw,v;\lambda )={}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda )+\mathrm{i}{}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda )$,
	${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(c,\alpha -\text{β})}(u,v,w;\lambda )=\text{β}!{\left(\frac{-1}{2}\right)}^{\text{β}}\sum _{r=0}^{\kappa }(\genfrac{}{}{0ex}{}{\kappa }{r}){}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa -r}^{(c,\alpha )}(u,v,w;\lambda )\phi (r,\text{β},-\lambda )$,
	${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda )=\kappa !\sum _{r=0}^{\left[\frac{\kappa }{\sigma }\right]}\frac{{\mathcal{E}}_{\kappa -\sigma r}^{(s,\alpha )}(u,w;\lambda )v^r}{r!(\kappa -\sigma r)!}$,
	${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda )=\exp \left(v{D}_u^{\sigma }\right)\left\{{\mathcal{E}}_{\kappa }^{(c,\alpha )}\right(u,w;\lambda \left)\right\}.$

Table 2. Results of the parametric kind 2-variable Hermite Kampé de Fériet-Eulerian-type polynomials.

Generating functions	${\left(\frac{1-\nu }{\lambda {\mathrm{e}}^{\tau }-\nu }\right)}^{\alpha }{\mathrm{e}}^{u\tau +v{\tau }^2}\cos \left(w\tau \right)=\sum _{\kappa =0}^{\mathrm{\infty }}{}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda ,\nu )\frac{{\tau }^{\kappa }}{\kappa !},$
	${\left(\frac{1-\nu }{\lambda {\mathrm{e}}^{\tau }-\nu }\right)}^{\alpha }{\mathrm{e}}^{u\tau +v{\tau }^2}\sin \left(w\tau \right)=\sum _{\kappa =0}^{\mathrm{\infty }}{}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda ,\nu )\frac{{\tau }^{\kappa }}{\kappa !}.$
Series	${}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda ,\nu )=\sum _{r=0}^{\kappa }(\genfrac{}{}{0ex}{}{\kappa }{r}){\mathbb{H}}_r^{\left(\alpha \right)}(\lambda ,\nu ){\mathcal{H}}_{\kappa -r}^{(c,2)}(u,v,w)$,
representations	${}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda ,\nu )=\sum _{r=0}^{\kappa }(\genfrac{}{}{0ex}{}{\kappa }{r}){\mathbb{H}}_r^{\left(\alpha \right)}(\lambda ,\nu ){\mathcal{H}}_{\kappa -r}^{(s,2)}(u,v,w)$.
Multiplicative and	${\hat{M}}_{{}_{{\mathcal{H}}^{\left(2\right)}}\mathbb{H}c}=u+2v{D}_u-\frac{\alpha \lambda {\mathrm{e}}^{D_u}}{\lambda {\mathrm{e}}^{D_u}-\nu }-w\tan \left(w{D}_u\right);{\hat{P}}_{{}_{{\mathcal{H}}^{\left(2\right)}}\mathbb{H}c}=D_u,$
derivative operators	${\hat{M}}_{{}_{{\mathcal{H}}^{\left(2\right)}}\mathbb{H}s}=u+2v{D}_u-\frac{\alpha \lambda {\mathrm{e}}^{D_u}}{\lambda {\mathrm{e}}^{D_u}-\nu }+w\cot \left(w{D}_u\right);{\hat{P}}_{{}_{{\mathcal{H}}^{\left(2\right)}}\mathbb{H}s}=D_u.$
Differential equations	$(u{D}_u+2v{D}_u^2-\frac{\alpha \lambda {\mathrm{e}}^{D_u}}{\lambda {\mathrm{e}}^{D_u}-\nu }{D}_u-w\tan (w{D}_u)D_u-\kappa ){}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda ,\nu )=0,$
	$(u{D}_u+2v{D}_u^2-\frac{\alpha \lambda {\mathrm{e}}^{D_u}}{\lambda {\mathrm{e}}^{D_u}-\nu }{D}_u+w\cot (w{D}_u)D_u-\kappa ){}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda ,\nu )=0$.
Some useful	${}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda ,\nu )=\kappa !\sum _{r=0}^{\left[\frac{\kappa }{2}\right]}\frac{{\mathbb{H}}_{\kappa -2r}^{(c,\alpha )}(u,w;\lambda ,\nu )v^r}{(\kappa -2r)!r!}$,
Formulae	${}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda ,\nu )=\kappa !\sum _{r=0}^{\left[\frac{\kappa }{2}\right]}\frac{{\mathbb{H}}_{\kappa -2r}^{(s,\alpha )}(u,w;\lambda ,\nu )v^r}{(\kappa -2r)!r!}$,
	${}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(s,\alpha )}(u+z,v,w;\lambda ,\nu )=\sum _{r=0}^{\kappa }(\genfrac{}{}{0ex}{}{\kappa }{r}){\mathbb{H}}_{\kappa -r}^{(s,\alpha )}(z,w;\lambda ,\nu ){\mathcal{H}}_r^{\left(2\right)}(u,v)$,
	${}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(s,\alpha +\text{β})}(u+x,v,w+y;\lambda ,\nu )$
	$=\sum _{r=0}^{\kappa }(\genfrac{}{}{0ex}{}{\kappa }{r})\left[{}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa -r}^{(s,\alpha )}\right(u,v,w;\lambda ,\nu \left){\mathbb{H}}_r^{(c,\text{β})}\right(x,y;\lambda ,\nu )+{\mathbb{H}}_{\kappa -r}^{(s,\alpha )}(u,y;\lambda ,\nu \left){}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_r^{(c,\text{β})}\right(x,v,w;\lambda ,\nu \left)\right]$,
	${}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{\left(\alpha \right)}(u+iw,v;\lambda ,\nu )={}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda ,\nu )+\mathrm{i}{}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda ,\nu )$,
	${}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(c,\alpha -\text{β})}(u,v,w;\lambda ,\nu )=\text{β}!{\left(\frac{\nu }{1-\nu }\right)}^{\text{β}}\sum _{r=0}^{\kappa }(\genfrac{}{}{0ex}{}{\kappa }{r}){}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa -r}^{(c,\alpha )}(u,v,w;\lambda ,\nu )\phi (r,\text{β},\frac{\lambda }{\nu })$,
	${}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(s,\alpha )}(u,v,w;\lambda ,\nu )={\left(\frac{\nu -1}{2\nu }\right)}^{\alpha }\kappa !\sum _{r=0}^{\left[\frac{\kappa }{2}\right]}\frac{{\mathcal{E}}_{\kappa -2r}^{(s,\alpha )}(u,w;-\frac{\lambda }{\nu })v^r}{r!(\kappa -2r)!}$,
	${}_{{\mathcal{H}}^{\left(2\right)}}{\mathbb{H}}_{\kappa }^{(c,\alpha )}(u,v,w;\lambda ,\nu )=\exp \left(v{D}_u^2\right)\left\{{\mathbb{H}}_{\kappa }^{(c,\alpha )}\right(u,w;\lambda ,\nu \left)\right\}.$

Figure 1. Graph of the PKGHAEP ${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(c,\alpha )}(u,5,6;\lambda )$.

Table 3. Real and complex zeros of the PKGHAEP ${}_{{\mathit{G}}^{\left(3\right)}}{\mathcal{E}}_{\kappa }^{(c,4)}(u,5,6;2)$.

κ	Real zeros	Complex zeros
1	2.66667	–
2	$-3.40696,8.74029$	–
3	$-7.98745,2.94061,13.0468$	–
4	$-12.2585,0.304016,5.61458,17.0066$,	–
5	$-16.3761,-2.22546,4.26241,$	–
	6.79514, 20.8773
6	$-20.4056,-4.56172,1.98061,24.7206$	$7.13307-1.5258i,7.13307+1.5258i$
7	$-24.3802,-6.74644,-0.0745285,$	$8.13847-2.64017i,8.13847+2.64017i$
	5.03761, 28.5533
8	$-28.3184,-8.81144,-2.02027,$	$8.90755-3.77884i,8.90755+3.77884i,$
	$3.21645,7.07077,32.3811$
9	$-32.2315,-10.7794,-3.87898,1.45178,$	$9.63565-4.93371i,9.63565+4.93371i$
	$5.87933,8.08102,36.2064$
10	$-36.1264,-12.6672,-5.66131,$	$8.17674-0.690615i,8.17674+0.690615i,$
	$-0.273646,4.35813,40.0302$	$10.3267-6.08457i,10.3267+6.08457i$

Figure 2. Graphs of ${}_{{\mathit{G}}^{\left(3\right)}}{\mathcal{E}}_{30}^{(c,4)}(u,5,6;2)$ (top) and ${}_{{\mathit{G}}^{\left(3\right)}}{\mathcal{E}}_{20}^{(c,4)}(u,5,6;2)$ (bottom).

Figure 3. Structure of real zeros of ${}_{{\mathit{G}}^{\left(3\right)}}{\mathcal{E}}_{\kappa }^{(c,4)}(u,5,6;2)$.

Figure 4. Stacking structure zeros of ${}_{{\mathit{G}}^{\left(3\right)}}{\mathcal{E}}_{\kappa }^{(c,4)}(u,5,6;2)$.

Figure 5. Graph of the PKGHAEP ${}_{{\mathit{G}}^{\left(\sigma \right)}}{\mathcal{E}}_{\kappa }^{(c,\alpha )}(u,5,6;\lambda )$.

Table 4. Real and complex zeros of the PKGHAEP ${}_{{\mathit{G}}^{\left(3\right)}}{\mathcal{E}}_{\kappa }^{(s,4)}(u,5,6;2)$.

κ	Real zeros	Complex zeros
1	–	–
2	2.66667	–
3	$-0.92344,6.25678$	–
4	$-3.9115,3.46326,8.44823$,	–
5	$-6.63887,1.24086,$	–
	5.97567, 10.089
6	$-9.18689,-1.02348,5.02073,$	–
	$6.90257,11.6204$
7	$-11.6082,-3.13291,$	$7.43057-2.07924i,7.43057+2.07924i$
	2.60609, 13.2739
8	$-13.9358,-5.13089,0.738042,$	$8.39279-3.43901i,8.39279+3.43901i,$
	$5.17584,15.0339$
9	$-16.192,-7.03508,-1.08619,$	$9.22078-4.75189i,9.22078+4.75189i$
	$3.85315,6.50744,16.8444$
10	$-18.3923,-8.85965,-2.83742,$	$6.65726-1.18058i,6.65726+1.18058i,$
	$2.08858,18.6803$	$10.003-6.00447i,10.003+6.00447i$

Figure 6. Graphs of ${}_{{\mathit{G}}^{\left(3\right)}}{\mathcal{E}}_{30}^{(s,4)}(u,5,6;2)$ (top) and ${}_{{\mathit{G}}^{\left(3\right)}}{\mathcal{E}}_{20}^{(s,4)}(u,5,6;2)$ (bottom).

Figure 7. Structure of real zeros of ${}_{{\mathit{G}}^{\left(3\right)}}{\mathcal{E}}_{\kappa }^{(s,4)}(u,5,6;2)$.

Figure 8. Stacking structure zeros of ${}_{{\mathit{G}}^{\left(3\right)}}{\mathcal{E}}_{\kappa }^{(s,4)}(u,5,6;2)$.