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1. A physicist's guide to the solution of Kummer's equation and confluent hypergeometric functions

   https://doi.org/10.5488/CMP.25.33203  

   Mathews, W. N.; Esrick, M. A.; Teoh, Z. Y.; Freericks, J. K. (January 2022, Condensed Matter Physics)

   The confluent hypergeometric equation, also known as Kummer's equation, is one of the most important differential equations in physics, chemistry, and engineering. Its two power series solutions are the Kummer function, M(a,b,z), often referred to as the confluent hypergeometric function of the first kind, and M ≡ z1-bM(1+a-b, 2-b,z), where a and b are parameters that appear in the differential equation. A third function, the Tricomi function, U(a,b,z), sometimes referred to as the confluent hypergeometric function of the second kind, is also a solution of the confluent hypergeometric equation that is routinely used. Contrary to common procedure, all three of these functions (and more) must be considered in a search for the two linearly independent solutions of the confluent hypergeometric equation. There are situations, when a, b, and a - b are integers, where one of these functions is not defined, or two of the functions are not linearly independent, or one of the linearly independent solutions of the differential equation is different from these three functions. Many of these special cases correspond precisely to cases needed to solve problems in physics. This leads to significant confusion about how to work with confluent hypergeometric equations, in spite of authoritative references such as the NIST Digital Library of Mathematical Functions. Here, we carefully describe all of the different cases one has to consider and what the explicit formulas are for the two linearly independent solutions of the confluent hypergeometric equation. The procedure to properly solve the confluent hypergeometric equation is summarized in a convenient table. As an example, we use these solutions to study the bound states of the hydrogenic atom, correcting the standard treatment in textbooks. We also briefly consider the cutoff Coulomb potential. We hope that this guide will aid physicists to properly solve problems that involve the confluent hypergeometric differential equation. 

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2. Determinant of F_p-hypergeometric solutions under ample reduction

   Alexander Varchenko (January 2022, Contemporary mathematics)

   We consider the KZ differential equations over C in the case, when the hypergeometric solutions are one-dimensional integrals. We also consider the same differential equations over a finite field F_p. We study the polynomial solutions of these differential equations over F_p, constructed in a previous work joint with V. Schechtman and called the F_p-hypergeometric solutions. The dimension of the space of F_p-hypergeometric solutions depends on the prime number p. We say that the KZ equations have ample reduction for a prime p, if the dimension of the space of F_p-hypergeometric solutions is maximal possible, that is, equal to the dimension of the space of solutions of the corresponding KZ equations over C. Under the assumption of ample reduction, we prove a determinant formula for the matrix of coordinates of basis F_p-hypergeometric solutions. The formula is analogous to the corresponding formula for the determinant of the matrix of coordinates of basis complex hypergeometric solutions, in which binomials (z_i−z_j)^{M_i+M_j} are replaced with (z_i−z_j)^{Mi+Mj−p} and the Euler gamma function Γ(x) is replaced with a suitable F_p-analog defined on F_p 

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3. Hypergeometric Integrals Modulo p and Hasse–Witt Matrices

   https://doi.org/10.1007/s40598-020-00168-2  

   Slinkin, Alexey; Varchenko, Alexander (January 2020, Arnold Mathematical Journal)

   null (Ed.)

   We consider the KZ differential equations over C in the case, when the hypergeometric solutions are one-dimensional integrals.We also consider the same differential equations over a finite field F_p. We study the space of polynomial solutions of these differential equations over F_p, constructed in a previous work by Schechtman and the second author. Using Hasse–Witt matrices, we identify the space of these polynomial solutions over F_p with the space dual to a certain subspace of regular differentials on an associated curve. We also relate these polynomial solutions over F_p and the hypergeometric solutions over C. 

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4. Continuum Energy Eigenstates via the Factorization Method

   https://doi.org/10.3390/sym15040797  

   Freericks, James K.; Mathews Jr., Wesley N. (April 2023, Symmetry)

   The factorization method was introduced by Schrödinger in 1940. Its use in bound-state problems is widely known, including in supersymmetric quantum mechanics; one can create a factorization chain, which simultaneously solves a sequence of auxiliary Hamiltonians that share common eigenvalues with their adjacent Hamiltonians in the chain, except for the lowest eigenvalue. In this work, we generalize the factorization method to continuum energy eigenstates. Here, one does not generically have a factorization chain—instead all energies are solved using a “single-shot factorization”, enabled by writing the superpotential in a form that includes the logarithmic derivative of a confluent hypergeometric function. The single-shot factorization approach is an alternative to the conventional method of “deriving a differential equation and looking up its solution”, but it does require some working knowledge of confluent hypergeometric functions. This can also be viewed as a method for solving the Ricatti equation needed to construct the superpotential. 

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5. A local discontinuous Galerkin method for nonlinear parabolic SPDEs

   https://doi.org/10.1051/m2an/2020026  

   Li, Yunzhang; Shu, Chi-Wang; Tang, Shanjian (January 2021, ESAIM: Mathematical Modelling and Numerical Analysis)

   null (Ed.)

   In this paper, we propose a local discontinuous Galerkin (LDG) method for nonlinear and possibly degenerate parabolic stochastic partial differential equations, which is a high-order numerical scheme. It extends the discontinuous Galerkin (DG) method for purely hyperbolic equations to parabolic equations and shares with the DG method its advantage and flexibility. We prove the L 2 -stability of the numerical scheme for fully nonlinear equations. Optimal error estimates ( O ( h (k+1) )) for smooth solutions of semi-linear stochastic equations is shown if polynomials of degree k are used. We use an explicit derivative-free order 1.5 time discretization scheme to solve the matrix-valued stochastic ordinary differential equations derived from the spatial discretization. Numerical examples are given to display the performance of the LDG method. 

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