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.

1.
L.
Infeld
and
T. E.
Hull
,
Rev. Mod. Phys.
23
,
21
(
1951
).
2.
This holds for the four types denoted as A‐D by Infeld and Hull; the other two types (E,F) are transformable into A‐D.
3.
To do this, we introduce a spurious variable, say φ, and and consider functions Fm(x,φ)≡Gm(φ)Ym(x) such that
k(x,m+1)⋅[Gm(φ)Ym(x)]≡K(x)(∂/∂φ)[Gm(φ)Ym(x)]
. Variations on this procedure occur in each one of the sections below. See, in this connection, the similar steps taken by
L.
Weisner
,
Pacific J. Math.
5
,
1033
(
1955
).
4.
W. Miller, Mem. Am. Math. Soc. No. 50 (1964).
5.
Higher Transcendental Functions, (McGraw‐Hill Book Company, Inc., New York, 1953), Vol. II. Equations (21a) and (21b) appear on p. 100.
6.
Reference 5, p. 45.
7.
Reference 5, p. 101.
8.
Reference 5, p. 102.
9.
Reference 5, p. 174ff.
10.
Reference 5, p. 183ff.
11.
E. D. Rainville, Special Functions (Macmillan Company, New York, 1960), p. 280.
12.
E. P. Wigner, Group Theory and its Application to Quantum Mechanics, Academic Press Inc., New York, 1959), p. 216.
13.
Reference 5, p. 194.
14.
C. Truesdell, “An Essay Toward a Unified Theory of Special Functions,” Ann. Math. Studies, No. 18, (Princeton University Press, Princeton, New Jersey, 1948).
This content is only available via PDF.
You do not currently have access to this content.