Table Of ContentTABLES OF
LAGUERRE POLYNOMIALS
AND
FUNCTIONS
V. S. AlZENSHTADT, V. I. KRYLOV
and
A. S. METEL'SKII
Translated by
PRASENJIT BASU
PERGAMON PRESS
OXFORD LONDON · EDINBURGH · NEW YORK
TORONTO · PARIS · FRANKFURT
Pergamon Press Ltd., Headington Hill Hall, Oxford
4 & 5 Fitzroy Square, London W. 1
Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1
Pergamon Press Inc., 44-01 21st Street, Long Island City, New York 11101
Pergamon of Canada, Ltd., 6 Adelaide Street East, Toronto, Ontario
Pergamon Press S.A.R.L., 24 rue des Écoles, Paris 5e
Pergamon Press GmbH, Kaiserstrasse 75, Frankfurt-am-Main
Copyright © 1966
Pergamon Press Ltd.
First English edition 1966
This is a translation of the original volume TaOAUvfri MHO-
ζοΗΛβηοβ u φυΗκυ,ν,ΰ Jlnzeppa (Tablitsy mnogochlenov
i funktsii Liagerra) published in 1963 by the Academy of Sciences
of the B.S.S.R., Minsk
Library of Congress Catalog Card No. 66-14492
Printed in Poland
2416/66
PREFACE
LAGUERRE polynomials and functions are widely used in many problems of mathe-
matical physics and quantum mechanics, for example, in the integration of Helm-
holtz's equation in paraboloidal coordinates, in the theory of the propagation of
electromagnetic oscillations along long lines, in the solution of Schrödinger's equa-
tion for hydrogen-like atoms, etc., as well as in the expansion in series of an arbitrary
function in the interval (0, oo).
The tables contain the values of Laguerre polynomials and Laguerre functions
for/i = 2,3,...,7; 5 = 0(0.1) 1; x = 0(0.1) 10(0.2)30 and the zeros and coefficients of
the polynomials for A* = 2(1) 10 and 5 = 0(0.05) 1.
The book is intended for workers at computer centres, research institutes and engi-
neering organizations.
Vll
INTRODUCTION
Laguerre polynomials and their properties
Generalized Chebyshev-Laguerre polynomials Ls (oc) occur in physics in connec-
n
tion with the solution of the second-order linear differential equation
xy" + (s+l-x)y' + ny = 0.
We shall define the polynomial Ls (x) by its expansion in powers of x.
n
Lsn(x) = xn- ^-(H+S)XW-1+ ~~ (/i + s)(tt + s-l)xn~2...
= Σ (j^^Än + s)(n + s-A)...(n + s-k + \)x"-k. (1)
*=o k\(n-k)\
Also L;(x) = (-l)"x-Jexj-(xi+ne"). (2)
dx
For 5 = 0 it is usual to write L (x) instead of L° (x).
n n
For brevity the polynomials Ls (x) are called Laguerre polynomials everywhere
n
below.
The orthogonality property
For s> — 1 the polynomials Ls (x) (H = 0, 1,2, ...) from an orthogonal system
n
on the semi-axis (0, oo) with the weight function XSQ~X
jxsc-xLs (x)Ls (x)dx = 0, ιηΦη. (3)
n m
o
The norm of Ls (x) is given by the equation
n
Jxse-x[Ls (x)Yax = n\r(s + n + \). (4)
n
o
Recurrence relations
The following relations, by means of which the parameters n and s can be altered
by a unit, are satisfied by the polynomials Ls (x).
n
xLs(x) = L^ (x) + (2M + s+l)L^(x) + n(n + s)L^ (x), (5)
n 1 1
IX
x Tables of Laguerre Polynomials and Functions
xL^Hx) = (n + s + l)L 5(x) + L^ (x) = (x-n)L s(x)-n(A2 + )L^ (x), (6)
n 1 n 5 1
V- \x) = V {x) + nV . (x) = X-^-~ Ls (x)-~Ls (x). (7)
n n x n n + t
n+s n+s
Generating functions and series in the polynomials Ls (x)
n
(l + tys-1cxp(^)=YLs (x)~ |ί|<1, (8)
r n
\l + i/ „=o n\
00 fn
(ί-ίΥ^ρ(χΟ=ΣΚ"'(χ)- \t\<l, (9)
n=o n\
oo (—\Ytn
Σ — ^ - f - — ^ nW = e'(xO" is/s(2VxO, s>-l. (10)
„to n!/\n + s + l)
For 5 = 0
oo C_J\n^n
(n!)2
n = 0
Derivatives and indefinite integrals
-L^(x) = πί^ΐ l(x) = -[(H + s)L;_ (x) + L s(x)] ,
t n
dx x
x
JLXOdi = ~[L-i(x)r =^L s W + L^-)J (11)
n o n +1 (
0
d
^L„s(x) + -i- L s (x)l = ax),
did i n +1
dm
~[xsL„s(x)]=( + s)(« + s-l)...(n+5-m + l)x s-mLrm(x), (12)
M
f^[e-VL„s(x)]=(-ire-V-mL s^(x), (13)
n
dx
X
Γ(η + α + β +1 ) j (x - 0*-lt°L°„(t) dt
o
= /-(iH-a+l)r(0)jc"+'L'+'(x), Rea>-1, Rej?>0, (14)
fe-'Li (i)di = e-Jt[L s(x) + Ls _ (x)]. (15)
n n n n l
Tables of Laguerre Polynomials and Functions
XI
Asymptotic representations of the polynomials Ls (x)
n
For real values of s, fixed x>0, the following equation is true uniformly on any
segment ε<χ^ζω (0<ε<ω<οο)
(—^ Ι 5(χ) = π-^^χ-^-ν5-^08[2(πχ)^^5π-ΐπ] + 0(^5-"). (16)
π
If s> — 1, we have uniformly on any segment 0<χ<ω< οο.
<ZlT e-*'x^x)-n^'^1)/,[(vx)*] + Q( *-») (17)
H
v = 4n + 2s + 2.
Laguerre functions
Laguerre functions are defined by the equation
Ψ' (χ) = *-*χχ**Ι%χ). (18)
η
Many of their properties can be obtained from the properties given above for the
polynomials L5 (x). For example, it can be easily verified that Ψ*(χ) is the solution
n
of the differential equation
/s+1 x s2\
(19)
For Ψ*(χ) (n = 0, 1, ...) we have the following orthogonality property
Ψ* (χ)Ψ° (χ)άχ = \ \ (20)
η ηι
o {nil (n+s+l) m = n)
The recurrence relations with respect to rt and s have the form
xΨΧΧ) = !P; ,(x) + (2n + s + 1) KW + » (« + s) ¥?- j(x), (21)
+
x^r1W = (« + s+l)K+^
+1
= (x - «) K(x) - n (« + s) V s_ (x), (22)
n t
χ*ΨΓ \x) = Ψ'. + n V._ ! = ^ ^1 ΨΙ- — Ψ° ,. (23)
η+
n + s n + s
Xll Tables of Laguerre Polynomials and Functions
Tabulation of the Laguerre polynomials and functions Ls (x) and Ψ*(χ)
n
The values of the Laguerre polynomials and functions Ls (x) and Ψ„(χ) depend
n
on the parameters n,x and s. The following values of n, x, s have been chosen
in order to keep the tables small.
Six values of n = 2(1) 7 were taken for the parameter «, the degree of the poly-
nomial Ls (x). The recurrence relations (5) and (21), with which the functions Ls (x)
n n+l
and Ψ* (χ) of the parameter n+\ can be calculated from the functions of the
+ί
two preceding values n and n — 1, allow the tables to be extended to values of n
greater than 7 by means of simple calculations. It should be borne in mind, however,
that such calculations often involve a loss in the number of correct figures and can
therefore usually be carried out only for a small number of steps. To see the need for
care it is enough to note that in equation (5) the coefficient of Ls (x) is a linear function
n
of «, while the coefficient of Ls - (x) is a polynomial of the second degree in n.
n X
The recurrence relations (6), (7), (22) and (23), with which the parameter s can be
changed by a unit, make it possible to confine the tabulation of Ls (x) and Ψ*(χ) to
n
any one interval of length 1 of the range of variation of s. The interval [0, 1] was
taken as such an interval. A set of eleven equidistant values of s (s = 0(0.1) 1) was
taken in it.
The recurrence equations given above may be used to calculate L5 (x) and Ψ*(χ),
n
if s lies outside [0, 1] and is of the form s = N/\0 (N an integer). The loss in accuracy
in such calculations at each step will, as a rule, be smaller than if equations (5)
and (21) are used.
The interval 0<x<30 is taken as the ränge of variation of x. This choice is made
because all zeros of the polynomial Ls (x) for 2<«<7 and 0<s<l lie inside this
n
interval. The following set of values of the argument is taken: x = 0(0.1) 10(0.2) 30.
The error in the values of Ls (x) and Ψ*(χ) given in the table does not exceed a unit
n
of the last decimal.
To extend the scope of calculations two additional tables are included in this
book, the coefficients and zeros of the Laguerre polynomials Ls (x). These are com-
n
piled for « = 2(1) 10, 5 = 0(0.05) 1 and all quantities are given to eight significant
figures.
If the tables of coefficients are used, Ls (x) can be found by using the expansion (1)
n
in powers of x by means of simple calculations.
If the tables of roots are used, Ls (x) can be found simply by using the well-known
n
expansion of Ls (x) into linear factors containing the zeros
n
Ls (x) = (x-x)(x-x )...(x-x ). (24)
n l 2 n
There is no restriction on the values of x in either case, and they may be real
or complex.
Tables of Laguerre Polynomials and Functions
xni
Arrangement and calculation of the tables
There are three tables in the book-tables of values of Laguerre polynomials
and functions, of the coefficients of the polynomials, and of their roots. The first table
consists of six parts arranged successively in the ascending order of the degree n.
Each of these parts, corresponding to a definite value of n, falls into 11 tables, ac-
cording to the value of the parameter s.
The tables of the coefficients
and of the zeros x of the Laguerre polynomials A: =1,2, ...,« are small in size and
k
their arrangement does not merit special comment. All the quantities given are
represented in the form xx ... x (p), where jq/O, and p is the decimal exponent
x 2 n
of the number. For example, 268537( — 2) = 0.268537 x 10-2. The initial zero and
decimal point are not given in the tables.
The tables were calculated for a wider range of values of the parameters n, s and x
(« = 2(1) 10, s = 0(0.05) 1, x = 0(0.1) 10(0.2)30(0.5)80) on electronic computers of
the Institute of Mathematics and Computer Technology of the Byelorussian Academy
of Sciences and the Computer Centre of the Academy of Sciences of the U.S.S.R.
The values of the Laguerre polynomials were found by using two different computing
schemes: by determining the zeros of the polynomials and using the expansion
(24), and by calculating the coefficents and finding the values of the polynomials
from Homer's scheme. In spite of the large loss of significant figures in the course
of calculations in some cases, satisfactory accuracy was obtained by comparing the
results obtained from different computers by the two methods described above.
Only a part of the tables retained at the Institute of Mathematics and Computer
Technology of the Byelorussian Academy of Sciences is reproduced here.
REFERENCES
1. G. SZEGÖ. Orthogonal polynomials. New York, 1959.
2. A. ERDÉLYI, W. MAGNUS, F. OBERHETTINGER and F. G. TRICOMI. Higher Transcendental Func-
tions, vol. II. New York, 1953.
3. V. I. SMIRNOV. A Course of Higher Mathematics, vol. Ill, part 2, Ch. 4. English translation
(from the original Russian) published by Pergamon Press, Oxford, 1964.
4. N. N. LEBEDEV. Special functions and their applications (Spetsial'nye funktsii i ikh prilozheniya).
GITTL, 1953.
5. V. S. AIZENSHTADT, V. I. KRYLOV and A. S. METEL'SKII. Tables for numerical Laplace transforms
00
and the calculation of integrals of the type j x8c~x f(x)dx (Tablitsy dlya chislennogo preobra-
o
zovaniya Laplaca i vychisleniya integralov vida). Byelorussian Academy of Sciences, Minsk, 1962.
xv
Tables of the values of
s
the polynomials L „(x) and the functions Ψ(*χ)
η
for «=2(1)7, 5=0(0.1)1, JC=0(0.1) 10(0.2)30
