In 1946, after Harry Bateman died, Whittaker was asked to recommend a mathematician who could start the task of publishing Bateman's manuscripts: the Bateman Manuscript Project.
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Erdélyi was primarily an expert in special functions, in particular, Lamé functions, hypergeometric functions and orthogonal polynomials.
It resulted in the eventual publication of five important reference volumes, under the editorship of Arthur Erdélyi.
The first definition was made by Meijer using a series; nowadays the accepted and more general definition is via a path integral in the complex plane, introduced in its full generality by Arthur Erdélyi in 1953.
The later Bateman Manuscript Project, under the editorship of Arthur Erdélyi, attempted to be encyclopedic, and came around the time when electronic computation was coming to the fore and tabulation ceased to be the main issue.
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From 1943 to 1945 and from 1948 to 1951 at the California Institute of Technology of Pasadena, he collaborated on the manual of special functions for the Bateman manuscript project, together with Arthur Erdélyi, Wilhelm Magnus and Fritz Oberhettinger.
Some of the mathematicians who have worked on orthogonal polynomials include Gábor Szegő, Sergei Bernstein, Naum Akhiezer, Arthur Erdélyi, Yakov Geronimus, Wolfgang Hahn, Theodore Seio Chihara, Mourad Ismail, Waleed Al-Salam, and Richard Askey.