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In this volume we will present some applications of special
functions in computer science. This largely consists of adaptations
of articles that have appeared in the literature . Here they are
presented in a format made accessible for the non-expert by
providing some context. The material on group representations and
Young tableaux is introductory in nature. However, the algebraic
approach of Chapter 2 is original to the authors and has not
appeared previously . Similarly, the material and approach based on
Appell states, so formulated, is presented here for the first time
. As in all volumes of this series, this one is suitable for
self-study by researchers . It is as well appropriate as a text for
a course or advanced seminar . The solutions are tackled with the
help of various analytical techniques, such as g- erating
functions, and probabilistic methods/insights appear regularly . An
interesting feature is that, as has been the case in classical
applications to physics, special functions arise- here in
complexity analysis. And, as in physics, their appearance indicates
an underlying Lie structure. Our primary audience is applied
mathematicians and theoretical computer scientists . We are quite
sure that pure mathematicians will find this volume interesting and
useful as well .
This series presents some tools of applied mathematics in the areas
of proba bility theory, operator calculus, representation theory,
and special functions used currently, and we expect more and more
in the future, for solving problems in math ematics, physics, and,
now, computer science. Much of the material is scattered throughout
available literature, however, we have nowhere found in accessible
form all of this material collected. The presentation of the
material is original with the authors. The presentation of
probability theory in connection with group represen tations is
new, this appears in Volume I. Then the applications to computer
science in Volume II are original as well. The approach found in
Volume III, which deals in large part with infinite-dimensional
representations of Lie algebras/Lie groups, is new as well, being
inspired by the desire to find a recursive method for calcu lating
group representations. One idea behind this is the possibility of
symbolic computation of the matrix elements. In this volume,
Representations and Probability Theory, we present an intro duction
to Lie algebras and Lie groups emphasizing the connections with
operator calculus, which we interpret through representations,
principally, the action of the Lie algebras on spaces of
polynomials. The main features are the connection with probability
theory via moment systems and the connection with the classical ele
mentary distributions via representation theory. The various
systems of polynomi als that arise are one of the most interesting
aspects of this study.
Introduction I. General remarks . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 1 II. Notations . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . 5 III. Lie
algebras: some basics . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8 Chapter 1 Operator calculus and Appell systems I. Boson calculus
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
II. Holomorphic canonical calculus . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18 III. Canonical Appell systems . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 23 Chapter 2 Representations of Lie groups I. Coordinates
on Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 II. Dual
representations . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 29 III. Matrix elements . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 37 IV. Induced representations and homogeneous
spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 40 General Appell systems Chapter 3 I. Convolution and
stochastic processes . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . 44 II. Stochastic
processes on Lie groups . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 46 III. Appell
systems on Lie groups . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
Chapter 4 Canonical systems in several variables I. Homogeneous
spaces and Cartan decompositions . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 54 II. Induced representation and
coherent states . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . 62 III. Orthogonal polynomials in several
variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 68 Chapter 5 Algebras with discrete spectrum I.
Calculus on groups: review of the theory . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 II.
Finite-difference algebra . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 85 III. q-HW algebra and basic hypergeometric functions . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . 89 IV. su2
and Krawtchouk polynomials . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 V. e2
and Lommel polynomials . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Chapter 6 Nilpotent and solvable algebras I. Heisenberg algebras .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . 113 II. Type-H
Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
118 Vll III. Upper-triangular matrices . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 125 IV. Affine and Euclidean algebras . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 127 Chapter 7 Hermitian symmetric spaces I. Basic
structures . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 131 II. Space of rectangular matrices . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 133 III. Space of skew-symmetric matrices . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 136 IV. Space of symmetric matrices . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 143 Chapter 8 Properties of matrix elements I. Addition
formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
147 II. Recurrences . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 148 III. Quotient representations and summation
formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 149 Chapter 9 Symbolic computations I. Computing the pi-matrices
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 153 II. Adjoint group . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 III.
Recursive computation of matrix elements . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . .
In this volume we will present some applications of special
functions in computer science. This largely consists of adaptations
of articles that have appeared in the literature . Here they are
presented in a format made accessible for the non-expert by
providing some context. The material on group representations and
Young tableaux is introductory in nature. However, the algebraic
approach of Chapter 2 is original to the authors and has not
appeared previously . Similarly, the material and approach based on
Appell states, so formulated, is presented here for the first time
. As in all volumes of this series, this one is suitable for
self-study by researchers . It is as well appropriate as a text for
a course or advanced seminar . The solutions are tackled with the
help of various analytical techniques, such as g- erating
functions, and probabilistic methods/insights appear regularly . An
interesting feature is that, as has been the case in classical
applications to physics, special functions arise- here in
complexity analysis. And, as in physics, their appearance indicates
an underlying Lie structure. Our primary audience is applied
mathematicians and theoretical computer scientists . We are quite
sure that pure mathematicians will find this volume interesting and
useful as well .
This series presents some tools of applied mathematics in the areas
of proba bility theory, operator calculus, representation theory,
and special functions used currently, and we expect more and more
in the future, for solving problems in math ematics, physics, and,
now, computer science. Much of the material is scattered throughout
available literature, however, we have nowhere found in accessible
form all of this material collected. The presentation of the
material is original with the authors. The presentation of
probability theory in connection with group represen tations is
new, this appears in Volume I. Then the applications to computer
science in Volume II are original as well. The approach found in
Volume III, which deals in large part with infinite-dimensional
representations of Lie algebras/Lie groups, is new as well, being
inspired by the desire to find a recursive method for calcu lating
group representations. One idea behind this is the possibility of
symbolic computation of the matrix elements. In this volume,
Representations and Probability Theory, we present an intro duction
to Lie algebras and Lie groups emphasizing the connections with
operator calculus, which we interpret through representations,
principally, the action of the Lie algebras on spaces of
polynomials. The main features are the connection with probability
theory via moment systems and the connection with the classical ele
mentary distributions via representation theory. The various
systems of polynomi als that arise are one of the most interesting
aspects of this study."
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