Abstract
Families of special functions, known from mathematical physics, are defined here by their recursion relations. The operators which raise and lower indices in these functions are considered as generators of a Lie algebra. The general element of the corresponding Lie group thus operates on the function in two ways: on the one hand it shifts the argument of the function; on the other hand it produces an infinite sum of functions (at the unchanged argument) with shifted indices. Equating the two results of the operation gives us ``addition theorems,'' hitherto derived by analytical methods. The present paper restricts itself to the study of 2- and 3-parameter Lie groups.
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