Christoffel-Darboux type identity

The classical Christoffel-Darboux identity for Hermite polynomials reads n∑k=0Hk(x)Hk(y)2kk!=12n+1n!Hn+1(x)Hn(y)−Hn(x)Hn+1(y)x−y. I am interested in a similar identity, where one of the indices is shifted by one, explicitly Fn(x,y):=n∑k=0Hk+1(x)Hk(y)2k+1(k+1)!=xn∑k=01k+1Hk(x)Hk(y)2kk!−n∑k=0kk+1Hk−1(x)Hk(y)2kk!, where the recursion relation Hk+1(x)=2xHk(x)−2kHk−1(x) was used. In the first term Christoffel-Darboux cannot be applied due to the 1/(k+1) prefactor and the second term is not quite Fn−1(y,x) … Read more

Spectrum of Kernel – Discrete orthogonal polynomials

Trying to solve a problem, I encounter a Kernel of the form K(m,n)=e−β4(m+n+1)22+m+n2√m!n!√πn−m[1Γ(−m/2)Γ(−n+12)−1Γ(−n/2)Γ(−m+12)] where m,n∈N and 0<q<1 a real parameter. I want to diagonalize this kernel and find its eigenvalues/eigenfunctions. It looks like an integrable kernel after having performed the Cristoffel-Darboux summation formula which has a general form like Kk(n,m)=√w(m)w(n)fk(n)fk−1(m)−fk(m)fk−1(n)n−m. and one should take the … Read more

Infinite tridiagonal matrices and a special class of totally positive sequences

Let y=(y1,y2,y3,…) be an infinite sequence of positive real numbers such that following N×N tridiagonal matrix T(y):=(1y10011y20011y30011⋱) is totally positive in the sense that all of its leading principal minors [1,…,n] for n≥1 are positive; here [j1,…,jn] denotes the principal minor of T whose row and columns sets are indexed by an (ordered) subset {j1<⋯<jn}⊂N. … Read more

Generalization of Frobenius formula involving Macdonald polynomials

Given a vector $\vec k=(k_1,k_2,\cdots)$ with $k_i$ are non-negative integers, the Newton polynomial $p_{\vec k}(x)$ is defined as \begin{equation} p_{\vec k}(x)=\prod_{j=1}^n p_j^{k_j}(x)~, \end{equation} where \begin{equation} p_j(x)=\sum_{i=1}^N x_i^j \end{equation} are power-sum symmetric polynomials. The Newton polynomials are homogeneous of degree $\ell=\sum_{j=1}^n j k_j$. They are related to the Schur polynomials $s_R(x)$ through the Frobenius formula \begin{equation} … Read more

Gegenbauer’s addition theorem for Jacobi polynomials

I have the following identity, ∫1−1dzj0(√x2+y2−2xyz)Pn(z)=2jn(x)jn(y), where x,y>0, Pn is a Legendre polynomial, and jn is a spherical Bessel function. This follows from Gegenbauer’s addition theorem (NIST 10.60.2), j0(√x2+y2−2xyz)=∑l≥0(2l+1)jl(x)jl(y)Pl(z). I am interested in the generalisation of the above to general Jacobi polynomials P(α,β)n, ∫1−1dzj0(√x2+y2−2xyz)(1−z)α(1+z)βP(α,β)n(z), with α,β>−1 and x,y>0. We have the following connection formula for … Read more

Legendre Q(n,x) function coefficients in terms of P(n,x) coefficients

Empirically, the Legendre functions of second kind, Qn(x), appear to be of form Qn(x)=Pn(x)2⋅ln(1+x1−x)+pn(x), with Pn(x) the Legendre polynomials of first kind and pn(x) some rational polynomial of degree n−1. This observation came up with my current reimplementation of Qn(x) for the Sage CAS. Probably it has to do with the Qn(x) satisfying the same … Read more

Orthogonal polynomials with respect to the lognormal distribution

I am currently doing some inspection on the orthogonal polynomials with respect to the lognormal distribution. Does anyone already work on that or know some cool references? All the best, Pierre-O. Answer Orthogonal polynomials with respect to the lognormal distribution go by the name of Stieltjes-􏰈Wigert polynomials. Two recent studies of their properties: Global Asymptotics … Read more

Reference for the exponential decay of Legendre coefficients

In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate. Longer: If $p_n$ is the $n$-th Legendre polynomial, and the Legendre expansion of a real function $f$ is $f(x) = \sum\limits_{n=0}^{\infty} \hat{f}(n) p_n (x)$, where $\hat{f} (n) = \int\limits_{-1}^1 p_n … Read more

Identities for Chebyshev polynomials of the second kind

While calculating an integral in a quantum mechanical problem by two different methods, I came across the following identity \sum_{k=0}^n\sum_{m=0}^{2k}(-2)^m\binom{2(n-k)}{n-k}\binom{2k}{k}\binom{2k}{m}\cos^m{\phi}U_m(\cos{\phi})=2^{2n}U_{2n}(\cos{\phi})P_n(\cos{2\phi}), where U_m(x) are Chebyshev polynomials of the second kind and P_n(x) is the Legendre polynomial. Is this identity known? Were similar identities considered in the literature? P.S. It seems this is a special m=0 case … Read more

Orthogonal Polynomials and Sturm Liouville operators

Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define L[u]=u″, then the n-th order Hermite polynomial satisfies LH_n (x) = -nH_n (x). The only proof of this miracle that I know of goes through solving the SL problem by power-series method, and on the other hand obtaining the orthogonal … Read more