Special Functions and the Theory of Group Representations by N. Vilenkin | (PDF) Free Download

Description

A standard scheme for a relation between special functions and group representation theory is the following: certain classes of special functions are interpreted as matrix elements of irreducible representations of a certain Lie group, and then properties of special functions are related to (and derived from) simple well-known facts of representation theory. The book combines the majority of known results in this direction. In particular, the author describes connections between the exponential functions and the additive group of real numbers (Fourier analysis), Legendre and Jacobi polynomials and representations of the group $SU(2)$, and the hypergeometric function and representations of the group $SL(2,R)$, as well as many other classes of special functions.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐good

⭐My background is physics, and we study many of the ODEs, such as Bessel functions, Legendre functions, Hermite functions, etc, andthe similarities between these functions, and the types of relations they obeyed, was always a curiousity that bothered me, and my fellow students. This book explains this phenomenon in a completely satisfying way, and the experience was wonderful. The book is very well written. In fact, the whole subject seems like a miracle, just showing how straightforward algebraic and geometrical considerations lead directly to ODEs whose solutions *all* must obey the same set of relationships, because they are, at bottom, relations that are inherited from group representation theory. This is one of my most prized books in my library. I cannot recommend this book enough, not only for the explanation of a long-standing source of mystery, but also because of the tremendous beauty in making a complete link between a geometric-algebraic structure, and the properties of associated differential equations.

⭐This book clarifies the reason for the existence of so many similar sets of orthogonalspecial functions with similar types of recurrence relations. They all stem from the relationshipbetween the differential operator and infinitesimal operators on lie groups, and their action on the representationsof the lie group. For any physicist or engineer who has ever wondered where all this similarstructure comes from, this is the explanation, told beautifully. Buy this book and enjoy!

⭐I’m doing research on representation theory of compact Lie groups, this book provides great details of constructing representations for some fundamental Lie groups and stress their connection to special functions.

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