Szmytkowski derived a certain integral with Gegenbauer polynomials. A natural generalization is to derive lookalike integrals with Jacobi polynomials. Six methods are treated to derive the first integral. The first method should be enough to prove the first integral, but by the other methods there arises remarkable formula such as par example a zero-balanced F3 Appell function which can be converted into a 2F1 hypergeometric function. Another three integrals complete the paper.
It is shown that the integrals of the Jacobi polynomials \begin{equation*}%\label{eq:Fn^J} \int_0^t (t-θ)^δP_n^{(α-\frac12,β-\frac12)}(\cos θ) \left(\sin \tfracθ2\right)^{2 α} \left(\cos \tfracθ2\right)^{2 β} dθ> 0 \end{equation*} for all $t \in (0,π]$ and $n \in \mathbb{N}$ if $δ\ge α+ 1$ for $α,β\in \mathbb{N}_0$ and $\max\{α,β\} > 0$. This proves a conjecture on the integral of the Gegenbauer polynomials in \cite{BCX} that implies the strictly positive definiteness of the function $θ\mapsto (t - θ)_+^δ$ on the unit sphere $\mathbb{S}^{d-1}$ for $δ\ge \lceil \frac{d}{2}\rceil$ and the Polyà …
We consider a generalisation of a definite integral involving the Bessel function of the first kind. It is shown that this integral can be expressed in terms of the Fox-Wright function ${}_pΨ_q(z)$ of one variable. Some consequences of this representation are explored by suitable choice of parameters. In addition, two closed-form evaluations of infinite series of the Fox-Wright function are deduced.
We consider direct and inverse Jacobi transforms with measures $dμ(t)=2^{2ρ}(\sinh t)^{2α+1}(\cosh t)^{2β+1}\,dt$ and $dσ(λ)=(2π)^{-1}\bigl|\frac{2^{ρ-iλ}Γ(α+1)Γ(iλ)} {Γ((ρ+iλ)/2)Γ((ρ+iλ)/2-β)}\bigr|^{-2}\,dλ$, respectively. We solve the following generalized Logan problem: to find \[ \infΛ((-1)^{m-1}f), \quad m\in \mathbb{N}, \] where $Λ(f)=\sup\,\{λ>0\colon f(λ)>0\}$ and the infimum is taken over all nontrivial even entire functions $f$ of exponential type that are Jacobi transforms of positive measures with supports on an interval. Here, if $m\ge 2$, then we additionally assume that $\int_{…
The definite integrals $ \int_{-1}^1x[P_ν(x)]^4\,\mathrm{d} x$ and $ \int_{0}^1x[P_ν(x)]^2\{[P_ν(x)]^2-[P_ν(-x)]^2\}\,\mathrm{d} x$ are evaluated in closed form, where $ P_ν$ stands for the Legendre function of degree $ ν\in\mathbb C$. Special cases of these integral formulae have appeared in arithmetic studies of automorphic Green's functions and Epstein zeta functions.
The aim of this paper is to apply generalized operators of fractional integration and differentiation involving Appell's function $F_{3}(:)$ due to Marichev-Saigo-Maeda (MSM), to the Jacobi type orthogonal polynomials. The results are expressed in terms of generalized hypergeometric function. Some of the interesting special cases of the main results also established.
In this work we shall apply the Bochner's theorem to prove certain combinations of Euler's q-exponentials are positive definite functions. Then we apply this positivity to prove curious inequalities for the Jacobi theta function $\vartheta_{4}$ and q-Gamma function $Γ_{q}$.
We present a method to obtain weight functions associated with linear and quadratic lattices that yield discrete orthogonality with respect to a quasi-definite moment functional of the orthogonal polynomial sequences in the Askey scheme, with the exception of the Jacobi, Bessel, Laguerre, and Hermite polynomials.
We study a family of orthogonal polynomials which satisfy (apart from a 3-term recurrence relation) an eigenvalue equation involving a third order differential operator of Dunkl-type. The orthogonality measure of these polynomials consists in the continuous measure of the little -1 Jacobi polynomials to which is added an arbitrary mass located at the point $x=0$, the middle of the orthogonality interval. This provides the first nontrivial example of Krall-type polynomials with a point mass inside the orthogonality interval. These polynomials can be obtained by a Geronimus transform of the litt…
On the definite integral of two confluent hypergeometric functions related to the Kampé de Fériet double series
通过求解相应的一阶非齐次微分方程，研究了Kampé de Fériet二重级数$F_{1:1;1}^{1:1;1}$.证明了$t^{β+l}M(\cdot;β;λt)M(\cdot;β;-λt)$在$t\in[0,T]$，$T\geq0$，$l=0,1,\ldots$，$\Reβ+l>-1$上的积分是函数$F_{1:1;1}^{1:1;1}$的线性组合.积分是一类所谓的库仑积分的推广，涉及规则库仑波函数。
We set out to build a framework for self-adjoint extension theory for powers of the Jacobi differential operator that does not make use of classical deficiency elements. Instead, we rely on simpler functions that capture the impact of these elements on extensions but are defined by boundary asymptotics. This new perspective makes calculations much more accessible and allows for a more nuanced analysis of the associated domains. The maximal domain for $n$-th composition of the Jacobi operator is characterized in terms of a smoothness condition for each derivative, and the endpoint behavior of f…
In this contribution we deal with a varying discrete Sobolev inner product involving the Jacobi weight. Our aim is to study the asymptotic properties of the corresponding orthogonal polynomials and the behavior of their zeros. We are interested in Mehler-Heine type formulae because they describe the essential differences from the point of view of the asymptotic behavior between these Sobolev orthogonal polynomials and the Jacobi ones. Moreover, this asymptotic behavior provides an approximation of the zeros of the Sobolev polynomials in terms of the zeros of other well-known special functions.…
Nonsymmetric Askey-Wilson polynomials are usually written as Laurent polynomials. We write them equivalently as 2-vector-valued symmetric Laurent polynomials. Then the Dunkl-Cherednik operator of which they are eigenfunctions, is represented as a 2x2 matrix-valued operator. As a new result made possible by this approach we obtain positive definiteness of the inner product in the orthogonality relations, under certain constraints on the parameters. A limit transition to nonsymmetric little q-Jacobi polynomials also becomes possible in this way. Nonsymmetric Jacobi polynomials are considered as …
We consider a certain definite integral involving the product of two classical hypergeometric functions having complicated arguments. We show in this paper the surprising fact that this integral does not depend on the parameters of the hypergeometric functions.
