Structural Formulas for Matrix-Valued Orthogonal Polynomials Related to $$2\times 2$$ Hypergeometric Operators

We give some structural formulas for the family of matrix-valued orthogonal polynomials of size $$2\times 2$$ 2 × 2 introduced by C. Calderón et al. in an earlier work, which are common eigenfunctions of a differential operator of hypergeometric type. Specifically, we give a Rodrigues formula that allows us to write this family of polynomials explicitly in terms of the classical Jacobi polynomials, and write, for the sequence of orthonormal polynomials, the three-term recurrence relation and the Christoffel–Darboux identity. We obtain a Pearson equation, which enables us to prove that the sequence of derivatives of the orthogonal polynomials is also orthogonal, and to compute a Rodrigues formula for these polynomials as well as a matrix-valued differential operator having these polynomials as eigenfunctions. We also describe the second-order differential operators of the algebra associated with the weight matrix.

ON THE MODULARITY OF SOLUTIONS TO CERTAIN DIFFERENTIAL EQUATIONS OF HYPERGEOMETRIC TYPE

We answer some questions in a paper by Kaneko and Koike [‘On modular forms arising from a differential equation of hypergeometric type’, Ramanujan J.7(1–3) (2003), 145–164] about the modularity of the solutions of a certain differential equation. In particular, we provide a number-theoretic explanation of why the modularity of the solutions occurs in some cases and does not occur in others. This also proves their conjecture on the completeness of the list of modular solutions after adding some missing cases.

On General Properties of Degenerate Systems of Second Order Partial Differential Equations of Hypergeometric Type

For the first time, the general properties of degenerate related hypergeometric systems such as Horn, Whittaker, Bessel and Laguerre are investigated together. The joint research allowed to reveal their various common properties and to establish a number of new degenerate related systems. They are all private cases of the common system offered by the authors for consideration. For the full study, it is important to classify its regular and irregular special curves and to identify the types of corresponding solutions. In this paper, they are implemented using simple rules. Special attention is paid to the construction of normal and regular solutions, because the solutions of all related degenerate systems such as Horn, Whittaker, Bessel and Laguerre near the irregular singularity on infinity relate to this species. Peculiarities of building normal-regular solutions by the Frobenius-Latysheva method are shown. All constructed normal-regular solutions are expressed through the function of Humbert  variables, which is the solution of degenerate hypergeometric system of Horn type. As an example, the cases  where, along with the application of the Frobenius-Latysheva method, the possibility of outputting new degenerate related systems is demonstrated.

On a Symmetric Generalization of Bivariate Sturm–Liouville Problems

A new class of partial differential equations having symmetric orthogonal solutions is presented. The general equation is presented and orthogonality is obtained using the Sturm–Liouville approach. Conditions on the polynomial coefficients to have admissible partial differential equations are given. The general case is analyzed in detail, providing orthogonality weight function, three-term recurrence relations for the monic orthogonal polynomial solutions, as well as explicit form of these monic orthogonal polynomial solutions, which are solutions of an admissible and potentially self-adjoint linear second-order partial differential equation of hypergeometric type.

The Asymptotic Expansion of a Function Introduced by L.L. Karasheva

The asymptotic expansion for x→±∞ of the entire function Fn,σ(x;μ)=∑k=0∞sin(nγk)sinγkxkk!Γ(μ−σk),γk=(k+1)π2n for μ>0, 0<σ<1 and n=1,2,… is considered. In the special case σ=α/(2n), with 0<α<1, this function was recently introduced by L.L. Karasheva (J. Math. Sciences, 250 (2020) 753–759) as a solution of a fractional-order partial differential equation. By expressing Fn,σ(x;μ) as a finite sum of Wright functions, we employ the standard asymptotics of integral functions of hypergeometric type to determine its asymptotic expansion. This was found to depend critically on the parameter σ (and to a lesser extent on the integer n). Numerical results are presented to illustrate the accuracy of the different expansions obtained.

Six-point conformal blocks in the snowflake channel

We compute d-dimensional scalar six-point conformal blocks in the two possible topologies allowed by the operator product expansion. Our computation is a simple application of the embedding space operator product expansion formalism developed recently. Scalar six-point conformal blocks in the comb channel have been determined not long ago, and we present here the first explicit computation of the scalar six-point conformal blocks in the remaining inequivalent topology. For obvious reason, we dub the other topology the snowflake channel. The scalar conformal blocks, with scalar external and exchange operators, are presented as a power series expansion in the conformal cross-ratios, where the coefficients of the power series are given as a double sum of the hypergeometric type. In the comb channel, the double sum is expressible as a product of two 3F2-hypergeometric functions. In the snowflake channel, the double sum is expressible as a Kampé de Fériet function where both sums are intertwined and cannot be factorized. We check our results by verifying their consistency under symmetries and by taking several limits reducing to known results, mostly to scalar five-point conformal blocks in arbitrary spacetime dimensions.