In this paper, some recursive relations of the derivatives of the gamma function Γ(α) and incomplete gamma function Γ(α, z) are obtained for complex number α ̸= 0,−1,−2, · · ·. Moreover, the recurrence relation of the derivative of the k−gamma function Γk(x) are also given. Thus, the partial derivative ∂ p+q ∂νp∂μq B(ν, μ) of the Beta function B(ν, μ) can also be represented for p, q = 0, 1, 2, ...

We investigate radial excitation of the quark-antiquark pair in ${\ensuremath{\pi}}^{0}$ meson and its effects on $\ensuremath{\gamma}{\ensuremath{\gamma}}^{*}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ transition form factor framework light cone perturbative QCD. The existing constraints wave function lowest Fock state $|q\overline{q}⟩$ allow a sizeable pair. construct for first radially e...

Intramembranous charge movement was measured in frog cut twitch fibers mounted in a double Vaseline-gap chamber with a TEA.Cl solution at 13-14 degrees C in the central pool. When a fiber was depolarized from a holding potential of -90 mV to a potential near -60 mV, the current from intramembranous charge movement was outward in direction and had an early, rapid component and a late, more slowl...

Abstract We show that independent and uniformly distributed sampling points are asymptotically as good optimal for the approximation of functions from Sobolev spaces $W_p^s(\varOmega )$ on bounded convex domains $\varOmega \subset{\mathbb{R}}^d$ in $L_q$-norm if $q<p$. More generally, we characterize quality arbitrary point sets $P\subset \varOmega $ via $L_\gamma (\varOmega )$-norm dist...

Abstract Let $\Gamma _{g}$ be the fundamental group of a closed connected orientable surface genus $g\geq 2$ . We develop new method for integrating over representation space $\mathbb {X}_{g,n}=\mathrm {Hom}(\Gamma _{g},S_{n})$ , where $S_{n}$ is symmetric permutations $\{1,\ldots ,n\}$ Equivalently, this all vertex-labeled, n -sheeted covering spaces g Given $\phi \in \mathbb {X}_{g,n}$ and $\...

We study harmonic analysis on cocommutative KPC-hyper-groups‎, which is a generalization of DJS-hypergroups‎, ‎introduced by Kalyuzhnyi‎, ‎Podkolzin and Chapovsky‎. ‎We prove that there is a relationship between‎ ‎the associated measures $mu$ and $gamma mu$‎, ‎where $mu$ is‎ ‎a Radon measure on KPC-hypergroup $Q$ and $gamma$ is a character on $Q$.

This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -\chi u(u+1)^{p-2}\nabla v +\xi u(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=\Delta v+\alpha u-\beta v, w+\gamma u-\delta w \end{cases} \end{align*} in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) smooth boundary $\partial\Ome...

Hyperbolic beta integrals are analogues of Euler’s beta integral in which the role of Euler’s gamma function is taken over by Ruijsenaars’ hyperbolic gamma function. They may be viewed as (q, q̃ )-bibasic analogues of the beta integral in which the two bases q and q̃ are interrelated by modular inversion, and they entail q-analogues of the beta integral for |q| = 1. The integrals under considerat...

Hyperbolic beta integrals are analogues of Euler’s beta integral in which the role of Euler’s gamma function is taken over by Ruijsenaars’ hyperbolic gamma function. They may be viewed as (q, q̃ )-bibasic analogues of the beta integral in which the two bases q and q̃ are interrelated by modular inversion, and they entail q-analogues of the beta integral for |q| = 1. The integrals under considerat...

We give an asymptotic expansion (the higher Stirling formula) and an infinite product representation (the Weierstrass product formula) of the Vignéras multiple gamma function by considering the classical limit of the multiple q-gamma function.