ACMD Seminar: Legendre functions: New formulas, identities, and transformations

Robert Maier
University of Arizona

Tuesday, August 3, 2021, 3:00 PM EDT (1:00 PM MDT)

A video of this talk is available to NIST staff in the Math channel on NISTube, which is accessible from the NIST internal home page.

Abstract: Legendre polynomials play a major role in applied and numerical mathematics, as do related functions: Legendre functions of the second kind, and associated Legendre functions of the first and second kinds.  Any of these function families can be viewed as orthogonal in a certain sense, which is why they frequently appear in numerical expansions of the solutions of partial differential equations (PDEs), in wave propagation problems and elsewhere.  Identities relating Legendre functions, and closed-form expressions for them, are therefore of value in mixed symbolic-numerical and purely symbolic computation.  In this talk, we review recent results on the transformation theory of these functions, which were obtained by considering them as solutions of ordinary differential equations (ODEs) of hypergeometric type, and changing the independent variable of the ODE in various ways.  This approach yielded many formulas not currently in the Digital Library of Mathematical Functions (DLMF), such as closed-form expressions for certain Legendre functions of fractional degree and order.  It also led to algebraic change-of-variables formulas, relating the value P(x) of a first-kind Legendre function to the value Q(y) of a second-kind one, where (x,y) traces out an algebraic curve in the plane.  Such identities extend well beyond the classical quadratic formula of Whipple.  The new approach also led to recurrence relations on the (fractional) degree and order of certain associated Legendre functions that may be useful in numerical expansions.

Bio: Robert Maier is professor of Mathematics and Physics at the University of Arizona, and adjunct professor of Mathematics at the University of Colorado Boulder.  His interests include special functions, symbolic computation, stochastic modelling, and applications of mathematics in bioinformatics and quantum computing.

Note: This talk will be recorded to provide access to NIST staff and associates who could not be present to the time of the seminar. The recording will be made available in the Math channel on NISTube, which is accessible only on the NIST internal network. This recording could be released to the public through a Freedom of Information Act (FOIA) request. Do not discuss or visually present any sensitive (CUI/PII/BII) material. Ensure that no inappropriate material or any minors are contained within the background of any recording. (To facilitate this, we request that cameras of attendees are muted except when asking questions.)

Host: Howard Cohl