Clebsch-Gordan-type linearisation relations for the products of Laguerre polynomials and hydrogen-like functions

Abstract

Two series of Clebsch-Gordan type linearisation relations are derived for the most general product of the Laguerre polynomials, Ln1alpha 1(u1x)Ln2alpha 2(u2x), which differ in orders, n, weights, alpha , and scaling multipliers, u. The general form and particular cases of coefficients in the expansion of the polynomial xkLn1alpha 1(u1x) . . . LnNalpha N(uNx) in terms of the Laguerre polynomials are established. The applications to hydrogen-like functions and Morse oscillators are indicated. Connection with an earlier Carlitz expansion, the technical links with the hyperspherical harmonics formalism and different approaches to the important Koornwinder's positivity theorems are discussed briefly.

Full text

J.
Phys. A: Math. Gen.
18
(1985) 1399-1417. Printed in Great Britain
Clebsch-Gordan-type linearisation relations for the products
of
Laguerre polynomials and hydrogen-like functions
A
W
Niukkanen
Laboratory of Molecular Spectroscopy and Quantum Chemistry,
V
I
Vernadsky Institute
of
Geochemistry and Analytical Chemistry, Academy
of
Sciences
of
the USSR, Moscow
117334, USSR
Received
5
August 1982, in final form 7 June 1984
Abstract.
Two series
of
Clebsch-Gordan type are derived
for
the most general product
of
the Laguerre polynomials,
L~;(u,x)L~;(u,x),
which differ in orders,
n,
weights,
a,
and
scaling multipliers,
U.
The general form and particular cases
of
coefficients in the expansion
of the polynomial
xkLz:(u,x).
.
.
Lz:(uNx)
in terms of the Laguerre polynomials are
established. The applications to hydrogen-like functions and Morse oscillators are indi-
cated. Connection with an earlier Carlitz expansion, the technical links with the hyper-
spherical harmonics formalism and different approaches to the important Koomwinder’s
positivity theorems are discussed briefly.
1.
Introduction
The Laguerre and Jacobi polynomials, which virtually cover all the classical orthogonal
polynomials, play an important role in various physical applications.
In
many cases,
the solutions of the Schrodinger equation for simple systems are expressed directly in
terms of such polynomials: for example, hydrogen-like functions via the Laguerre
polynomials, rotator functions via the Jacobi polynomials, etc. Since the Hermite and
Bessel polynomials are particular cases of the Laguerre polynomials, and the Legendre
and Gegenbauer polynomials are particular cases of the Jacobi polynomials, the
numbers of such examples may be easily extended.
The Laguerre and Jacobi polynomials also play an important role in approximate
variational solutions of complex many-electron systems, because basis functions in
variational methods are frequently connected with these two classes of special functions.
It is convenient (and, as a rule, such a procedure cannot be avoided in practice)
to represent the product of polynomials, p,(x)p,( x), arising in quantum mechanical
applications, as a linear combination of some functions
p;,
i.e. to use some linearisation
theorem. If
pk
polynomials of the same type, as in the initial product, are used as the
elements of such a linear combination, then the corresponding expansion is
of
Clebsch-
Gordan type (it is just this structure which is peculiar to the Clebsch-Gordan series
for spherical functions). Sometimes
it
is more suitable to use in linear combination
some functions
pi
which differ from
pk
We call such an expansion the (modified)
series of Clebsch-Gordan type.
Another important class of relations for classical polynomials is constituted by
addition theorems which either relate to an expansion off(x
+
y),
as in elliptic functions,
or
to an expansion off(g(x,,
.
.
.
,
x,)) where g(x,,
. . .
,
x,)
is an appropriate function
0305-4470/85/091399
+
19$02.25
@
1985 The Institute of Physics 1399

1400
A
W
Niukkanen
of some variables that are usually related to a distance function
on
a homogeneous
space. There are a number of addition formulae for the Jacobi and Laguerre poly-
nomials (Erdelyi 1953, Vilenkin 1965). Some new addition formulae for the Laguerre
polynomials were given by Koornwinder
(
1977) and Durand
(
1977).
As regards the linearisation theorems, they are rather numerous for the Jacobi
polynomials (Vilenkin 1965)
or,
equivalently, for the Wigner
D
functions (Varshalovich
et
al
1975). However, in the case of the Laguerre polynomials the linearisation
theorems, except for one general theorem of the modified type (Carlitz 1957)t, relate
to special cases rather than to a general case. For example, for a product
Lz;(
u,x)
Lz;(
u2x)
the following particular cases of modified linearisation theorems
have been considered: the case
a,
=
a2,
n,
=
n2
has been studied by Bailey (1936) (see
also equation 10.12 (42), Erdilyi 1953), and the case
a,
=
az,
U,
=
u2
by Howell (1937)$.
The usual (non-modified) Clebsch-Gordan type expansions have not been, to the
author's knowledge, considered
so
far in explicit form.
The need for the Laguerre polynomials in more general linearisation theorems is
implied by their importance in atomic and nuclear shell theories. One more reason is
that the hydrogen-like functions have been intensively advanced in recent years as
perspective basis functions for variational calculations of molecular electron wavefunc-
tions.
Some interesting mathematical problems arise in connection with the linearisation
relations§. For example, in particular cases the coefficients
C
in Clebsch-Gordan type
expansions satisfy some important inequalities. Very interesting results were found by
Koornwinder (1978). He showed that for integral
k,
I
Lk(
x)
LL(
x)
=
(-
1
)T,L::"-,(
x)
(1)
L",Ax)Lz((l
-A)x)=c ck(A)Lz+n-k(x)
(2)
I
where
C,
a
0,
and
k
with
Ck(
A
)
3
0
when
0
d
A
S
1
and
a
a
0.
These relations are useful for computational
purposes. For example, if
a
=
0
then
XkCk(A)
=
1
(set
x
=
0
in equation
(2)),
so
the
computation with these coefficients will be very stable for many problems. The
coefficients
Ck(A)
also have an interesting combinational meaning (Askey
et
a1
1978).
Since Koornwinder (1978) did not give explicit expressions for
C,
and
ck(h)
and his
original positivity proofs seem to be very cumbersome (especially for
C,(A)),
it would
be interesting to see whether the positivity of
C,
and
Ck(A)
follow directly from explicit
algebraic formulae. This problem is discussed briefly in
§
611.
t
This theorem is a particular case of more general expansion given in
9:
7.
This particular expansion has
been intensively studied in nuclear physics in recent years (the references are given in
8
7).
$
he expansion of the product
@(U,
c;
x)@(u',
c;
x),
which is equivalent to Howell's expansion, is presented
in equation
6.15
(29)
by Erdklyi (1953) with some misprints, as well as Howell's expansion (1937). The
correct forms of the expansions are given by Burchnall and Chaundy (1941) (relations
(72)
and
(98),
respectively).
I
The author is indebted to the referee for the comments which are used in the rest
of
this section.
/I
It
is shown that simple positivity proofs for
C,
and
Ck(A)
based
on
explicit algebraic expressions are,
actually, possible. However, we give in the following only sketches of the proofs rather than the proofs
themselves in full detail. The reason is that a number of other interesting properties of
Ck(A),
including
generating functions, addition and linearisation theorems, etc, are implied by
our
approach. The presentation
of these results would therefore lead to a conspicuous deviation from the initial objectives of the present
investigation. The corresponding results will be presented in a separate publication.

Linearisation relations for Laguerre polynomials
1401
2.
Expansion
of
x'Li(7x)
in terms
of
the Laguerre polynomials
Consider the expansion
Since ~~=(-l)~k!L;~(x), expansion (3) may be considered as a special form of
linearisation theorem for the Laguerre polynomials. Without a loss in generality, we
may confine ourselves to the case
(T
=
1,
because
Introducing the notation for the scalar product
(f;
cp),
=
Iom
dxx" exp(-x)f(x)cp(x)
and taking into account the orthogonality relation for the Laguerre polynomials
(Lz,
=
S(m,
n)T(a+ 1
+
n)/n!
(7)
we obtain
C?,:(k,
T)=
n!/T(a+
1
+n)
dxx'+k exp(-x)L:(x)Li(Tx).
(8)
Ip
Prior to using equation
(8)
for establishing the algebraic expression for
C(k,
T)
coefficients, we shall make two general remarks.
The first observation concerns 'selection rules'. The scalar product of a classical
polynomial (with 'its own' weight function) of degree
n
by any polynomial
pN
of
degree
N
is not zero only
if
n
s
N
(see
0
10.3 in Erdilyi (1953)), i.e. for example:
(L:,pN)u
$0
if
n
d
N.
(9)
~(k,
7)
=
n
!/r(a
+
1
+
n)(L:,
x~LP,(Tx)),
(10)
Presenting equation
(8)
in the form
we obtain in the case of integer
kaO
(the polynomiality condition for xkLi) the
evident selection rule
(11)
Osnsm+k.
If
T
=
1
(the case frequently arising in applications), there is an alternative expression:
C(k,
T)
=
n!/r(a
+
1
+
n(LP,,
x~-~+~L:)~.
(12)
This means, by virtue of equation (9), that
if
a
-
p
+
k
is non-negative integer (the
polynomiality condition for the second multiplier in equation (12)), then there exists
the supplementary selection rule and, then, in the case
a
-
p
+
k
<
m,
the number of
terms in the sum (3) is determined not by condition
(1
l), but obeys a stronger inequality
(13)
m-a+P-ksnsm+k

1402
A
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Niukkanen
In
particular, if
a
-
p
+
k
=
0,
then
m
n
m
+
k.
Thus, in the case of
k
and
a
-
p
integers, with fixed values of
m,
fl
and
k
in the left-hand side of equation
(3),
by
choosing a parameter
a
one may control the number of terms in the right-hand side
of
equation
(3).
Evidently, the least possible number of terms corresponds to the case
The second remark concerns the orthogonality relation and the sum rule for
C(
k,
T).
(~=p-k.
Applying the scaling transformations, one may easily show that
Substituting equation
(14)
into equation (4), we obtain
and, hence,
Note that in the case
k
=
1,2,.
.
.
,
the orthogonality relation links the coefficients
C(k,
T)
in expansion (4) of polynomial type
(k>
0)
with coefficients
C(-k,
1/~) of
the corresponding expansion of non-polynomial type.
In
the case
k=O
both sets of
coefficients in equation
(1
5)
pertain to polynomial expansions of the same type.
Multiplying both sides of equation
(3)
by the power x4 and using, one the one
hand, the expansion of type
(3)
for the expression arising
on
the left-hand side of the
equation
xk+‘Li(7x)
=c
Ck,r(k+q,
T,
t)L:(tx)
n
and,
on
the other hand, the expansion of the same type for the expressions arising
on
the right-hand side of the equation
we obtain, after obvious manipulations,
The ‘sum rule’ (16) can also be interpreted as an argument multiplication theorem
for
the function
C(
k,
T),
or
alternatively, as an addition theorem for the index
k.
One
may also easily show that the orthogonality relation (15) is a particular case of the
addition theorem (16).
3.
Algebraic representation for coefficients
C(k,
.r)-particular cases
In
order to obtain ‘explicit expressions’ for the coefficients
C,
we use the integral
representation equation
(8).
Writing down the Laguerre polynomials in equation
(8)
in the form of a hypergeometric series,
Fi,
we obtain the expression for
C(
k,
T)
as an

Linearisation relations for Laguerre polynomials
1403
Appell function,
F,t:
where
(a),
is the Pochhammer symbol.
the unit argument (see equation (16) in Niukkanen (1983)), we have
Using the reduction formula for the corresponding Lauricella function,
FA,
with
(a
+
1
)k(
p
+
I),
(-
k),
m!(a+
l),
+
1
+
k, k
+
1,
-
m
;
7
F:[
k+
1
-
n,
/3
+
1
C?P,(
k,
T)
=
Expression (1 8) is fallacious in some cases. Indeed, for the value
n
5
k
+
1, allowed
by selection rule (ll), the quantity (-k), in equation (18) assumes zero value, and
the series
F:
tends to infinity, since the parameter k
+
1
-
n
in
F:
is a non-positive
integer and, hence, the coefficients of
r'
in the sum
F:,
in the case
i
5
n
-
k, contain
zero denominators, (k
+
1
-
n)
;
therefore, the condition
i
s
m
does not lead to termina-
tion of the series before the 'dangerous denominators' appear. Consequently, in the
case of
n
2
k
+
1
expression
(
18) is formal, and to make it sensible it is necessary to
use some limiting transition
or
to apply some other calculation procedure that would
not result in the appearance of 'dangerous denominators'.
Since the hypergeometric series in equation (18) is a finite sum, it may be written
in 'inverse order' by reordering it in descending rather than ascending powers of
argument. Using for this purpose the general formula (35) from Niukkanen (1983),
we obtain the expression for the coefficient
C(
k,
T)
in terms of the Appell function,
F3:
Applying the reduction formula (17) from Niukkanen (1983), to the corresponding
Lauricella function
FB,
with the unit argument, we have
1
-m,-m-p,-k-min;
1/r
-m
-a
-
k,
-k-m
x
F:
Obviously, expression
(20)
is correct for any
n
such that
0
s
n
s
m
+
k. Really, in
spite of negative integer denominators, the negative integer numerators assure the
termination of the series before the diverging coefficients appear in the sum F:.
In
the case
T
=
1
the coefficient
C(
k,
T)
is
expressed in terms of
F:(
1).
In turn, any finite series
Fi(1)
is equivalent
to
a Clebsch-Gordan coefficient
(Smorodinsky and Shelepin 1972), which indicates an indirect link of the problem
under consideration with the linearisation theorem for spherical functions.
Let us consider some particular cases of expansion (4).
In
the case k
=
0
we obtain
the expansion of the Laguerre polynomial
L!(Tx)
over the Laguerre polynomials of
t
We use in equation
(17)
and hereafter the notation accepted in Niukkanen
(1983)).
@
indicates an empty
set.

1404
A
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Niukkanen
different weight with unit scaling multiplier
Li(
TX)
=
c
ci,:(o,
T)L:(X)
n
the corresponding function
Fi
in equation (20) being reduced to the function
F:
due
to cancellation of numerator and denominator parameters, i.e.
-m-p,-m+n;
1/r
-m-a
C?,Z(O,
7)
=-
(a
+
n
+
l)m-n(-m)nTmF:
m
!
If, in addition,
a
=
p
or
T
=
1,
then further reduction of the series
F:
takes place: in
the first case due to additional cancellation
of
numerator and denominator parameters,
and in the second case as
a
consequence
of
the Gauss theorem (Erdtlyi
1953,
equation
2.1 (14)).
As a result, we obtain, as particular cases of equation (21), two standard
expansions (Erdtlyi
1953,
equations
10.12 (40,39)):
Lp,(Tx)=c
n
CP,,p,(O,
7)L{(x)=C
,,
(P+m)i"(l-T)m-nLp(x) m-n (23)
and
Other cases of reduction of
F:
to
F:
in equation
(20),
which we shall not write down
in explicit form, correspond to
k
=
p,
k
=
-a
and
k
=
/3
-a,
which shows a particular
simplicity of expansions with functions of
x"Lz
form
on
the left- and/or right-hand
side
of
an expansion. The case
m
=
0
or
p
=
-m
corresponds to the expansion of an
exponential function over the Laguerre polynomials (in this case
F:(
1)
=
1
in equation
(20)).
In
some particular cases the series
F:(
1)
can be written as
a
simple
r
product
with the aid
of
the well known summation theorems (see
Q
4.4
in Erdtlyi
(1953)).
The
case
p
=
-2m
-
1
leads to expansion
of
the Bessel polynomial over the Laguerre
polynomials.
It is worth mentioning that there are alternative methods of obtaining equation
(20)
directly?. First, one can combine the known formulae
(23)
and
(24)
to
obtain an
expansion with coefficients
,F,(z)
where
z
=
T(T
-
l)-'.
Applying the analytical con-
tinuation formula
2
+
1
-
2-l
(see equation
2.10(4),
Erdtlyi
1953),
one can derive
equation
(22)
independently of
(20).
Differentiating equation
(22) k
times with respect
to
T,
we obtain, eventually, equation
(20).
Another approach is to use the explicit
algebraic expressions for Laguerre polynomials in equation
(8)
to obtain
C
as a double
sum. Expressing the resulting integrals in terms of the
r
functions and transforming
the double sum to a one-fold one with the help of the Gauss summation theorem$ for
*F,(l)
one can prove equation
(18)
directly. Moreover, equation
(18)
can be trans-
formed into equation
(20)
by ordering the sum
F
in descending rather than ascending
powers
of
the argument
T.
Therefore one can proceed without the use of the general
formulae (Niukkanen
1983)
in this relatively simple case. Nevertheless the
use
of the
reduction formulae may serve
as
a paradigm for more complicated cases (see
§
51,
where it allows
us
to save much calculational effort.
t
The author is indebted to the referee for this indication.
$
The
use
of
the Gauss theorem is, actually, equivalent to what is partially implied by 'reduction rules'.

Linearisation relations
for
Laguerre polynomials
1405
4.
Coefficients
C(k,
7):
physical and formal applications
Using the coefficients
C(k,
T)
of the particular types (22), (23) and (24), one may
easily obtain alternative forms of expansions of the general coefficients
C(k,
T)
in
equation
(20),
in terms of simpler functions.
For
this purpose the polynomial multipliers
in the integrand of equation
(8)
should be grouped in such a way that it would be
possible to use any two of the three particular expansions indicated in
0
3,
as well as
the orthonormality integral (7). Using,
for
example, in equation
(8),
the expansions
LZ(X)
=c
c;;,u+k(o,
l)L:+k(x)
r
LE(TX)=C c~,~'k(o, T)L:'k(X)
5
we obtain, with the help of equation (7),
By virtue of relations (24) and (22), equation (25) yields an expansion of
F:(
1/~) in
terms of the Gauss functions,
F:(
l/~).
In
the case
T
=
1
equation (25) results in one
of the Thomae relations between F:(l) series (Bailey 1935a) which can be used, in
particular, for transition to a more symmetric (by n,
a
and
rn,
/3
parameters) form for
F:(
1)
series. Another representation of similar type arises when the functions xkL:(x)
and L:(Tx)
in
equation
(8)
are expanded over the Laguerre polynomials, L:(x).
The coefficients
C(k,
T)
also allow a number of straightforward physical applica-
tions.
For
example, when the Morse oscillator eigenfunctions (Wallace 1976, Ephremov
1977)
or
the eigenfunctions of the Strum-Liouville equation for the Morse oscillator
(Ephremov 1978) are used as a basis set in the variational theory of oscillations of
polyatomic molecules, 'a great many matrix elements involving these eigenfunctions
must be evaluated. That theory would not be practical if analytic expressions for the
matrix elements could not be found' (Wallace 1976). The matrix elements in this quote
have the following form in Wallace's (1976) notation:
where
f(x)
=
1, x, xdldx. Analytical expressions for integrals (26) were given by
Wallace (1976), depending
on
the form of the operator
1;
as a combination of some
cumbersome sums. With the aid of coefficients
C(
k,
T)
one may obtain a much more
compact expression
for
integrals (26) both in the case
f
=
1,
x, xdldx, and for a wider
class
of'
operators
f
=
x'(d/dx)'. Indeed, using the differentiation formula for the
Laguerre polynomials and taking into account equation
(8),
we obtain
By virtue of equation (20) the integral
I
is expressed as a finite sum Fi(1).
calculating matrix elements with hydrogen-like functions,
H(
r):
The coefficients
C(k,
T)
also arise in the theory of molecular electronic states when

1406
A
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Niukkanen
where
Ylm(r)
is the spherical function. The case
w
=
l/n
corresponds to a hydrogen
atom, and
w
=
Z/n
to a hydrogen-like atom with nuclear charge,
2.
Introducing the
reduced density
pJi(r)
=
{Hulnl/l(r)@ Hw2n2/2(r)}lm $29)
where the symbol
{
@}
denotes the irreducible tensor product (Varshalovich
et
a1
1975),
one may easily show, for example, that the probability of an electrical 2'-pole transition
is associated with the integral
dr
{9y(r)@p;2(r)}oo
where
@,,,(r)
=
rfY/m(r)
is the regular solid harmonic. Taking into account that
(Varshalovich
et
a1
1975)
where
(aabp
(cy)
is
the Clebsch-Gordan coefficient, we obtain
Ti2=
(~T)"~H(/,,
12,
l)H(l,
1,
0)(2wl)'i(2w2)~2Zf2/(w1+w2)~~+~~+~+3
exp(
-
r)
~t;l+'
(
xl
r)
L~>+I
(x2
r)
(31)
I;2
=
dr
,./1+/2+/+3
1:
where
12
=
/I,
-
11,
11,
-
11
+
2,
. .
.
,
I,
+
1,
x,=l+v x*=l-v
v1
=
n,
-
1,
-
1
v
=
(
w,
-
w2)
(
w,
+
q-1
v,
=
n,
-
I,
-
1.
Generally, the integral
(31)
is expressed in terms of the Appell function,
F,,
For
nl,
+
nl,
transitions a more simple expression via the
C(
k,
T)
coefficient takes place:
i.e. in accordance with equation
(20),
the probability of any multipole transitions, in
which the principal quantum number does not change, is expressed in terms of the
Clausen function,
F:(
1).
Transforming a part of the polynomial multipliers in the integrand
(31)
with the
help of equation
(4),
21*+
1,2/1+1
(12+1-11+1, l)~;'i+'(x,r)
r/2+/-/,+l
21
+I
LY;
=
c
C">.fl
n
and taking into account the reduction formula (ErdClyi
1953,
equation
5.10
(3))

Linearisarion relations for Laguerre polynomials
1407
we obtain in the general case
By virtue of the selection rule
(13):
v2+l2-l1
-I-
1
s
ns
v2+12+Il+
1
the number
of
terms in the right-hand side of equation (32) is 21+3, i.e. increases
proportionally to the transition orbital moment. The formulae obtained in this section
provide a convenient generalisation of expressions
for
the probability of dipole elec-
trical transitions (see earlier works
on
the hydrogen atom cited by Condon and Shortley
(1935)) for the case of any multipole transitions.
It is worth noting that expansion (4) also proves to be very useful in calculations
of
the Fourier transforms of atomic orbitals
in
the
MO
LCAO
SCF
method (Niukkanen
1984a).
5.
Generalisation
for
the case
of
N
variables
The reduced density, equation (29), also enters, as a typical integrand structure, in
more complicated four-centre integrals with hydrogen-like functions in variational
calculations of molecular electronic structure. The calculation of these integrals is
simplified
if
the irreducible product of the functions
H,
appearing in equation (29),
is
expressed as a linear combination of functions
H.
With due respect to equations
(27) and (28) such a linearisation relation should have the form
{H,ln,,l(r)OHw*n2/*(r)}fm
=
1
Ql,lf2n:n(w1,
~2)~w~+w2,nlm(r)
n
where coefficients
Q(w,,
w2)
are related to the coefficients
R(w,,
w2)
in the expansion
+,+f*-f
21
+I
R
ixln
(
w
w2)
Lil?,'-
2(
w
,
+
w2)
r]
by
(33)
L,
,L
II
-
(2w
,
r
)
ti)?l:-
,
(
20,
r
)
=
n
f
Iff
(2~1+2~2)
Qili2n(wl,
~2)
=
(2w1)'1(2w2)'2~(119 12,
I)Rif2,:n(wl,
~2).
In
turn, relation (33) is an obvious generalisation of expansion (3) for the case of two
Laguerre polynomials in the left-hand side
of
the equation. Since general formulae
have a similar structure for any number of the Laguerre polynomials, we consider the
most general expansion
of
this type
(34)
tkL:;(xlt).
.
.
L;:(xNt)
=C
c:;
:$(k;
xlr..
.
,
xN;
x)t;(xt).
n
Obviously, the coefficients
R
in equation (33) are
a
particular case of the coefficients
C:
(35)
21
+1,21
+I,Zf+l
Rl,llfi2n(Ul,
02)
=
Cn,l-/,-l~n~-f~-,,n-f-l(l,
+
12-
I;
2~,,2~2; 2Wl+2w2).

1408
A
W
Niukkanen
General properties of the coefficients
C
in equation
(34)
may be easily obtained by
analogy with the reasoning given in
00
2
and 3.
In
particular, just as in the case of
equations
(4)
and
(5),
we have
Cz;:,;'$(k;
x,,
..
.
,
x,.,; x)
=
x-kC:;;;::;;(
k;
>,
X
.
.
.
,")
X
where
Transforming the integral representation
dtr"+k exp(-t)L:;(x,r).
.
.
n!
-
-
to a form similar to equation (lo), and for the case
x,
=
1
to a form similar to equation
(12), we obtain the following selection rule:
max[O,
n,
-
n,
-. .
.
-
n3-,
-
ns+l
-.
.
.-
nN
-a
+
a,
-
k]s
n
Using exactly the same reasoning as in
§
3, we arrive at the following two expressions
for coefficients
C
in
terms of the functions
'"F
(Niukkanen 1983), which are similar
to equations (18) and
(20),
respectivelyt
k+
n,
+.
.
.+
nN.
3
a+k+l,k+l;-n,
,...,
-nN;x,
,...,
x,.,
k-
n
+
1;
ai+
1,.
. .
,
aN
+
1
(37)
-1
,.,
I
n
-
n,
-.
.
.
-
nN
-
k; -n,,
-a,
-
n,,
.
.
.
,
-nN,
-aN
-
nN;
x,
,
.
. .
,
x,'
-a
-
n,
-.
.
.
-
nN
-
k,
-n,
-.
.
.
-
nN
-
k;
Q..
.i~
(38)
Generally, to calculate the coefficients
C,
recurrence relations and explicit expressions
for the functions
NF
given in Niukkanen (1983) may be used.
In
the special case of
the coefficients
C
in equation
(35),
in which the quantities
k,
a,
and
a
are interrelated
by a linear relation, it is expedient to use such a recurrence equation that would not
violate this relation, i.e. that would not involve in the recursion some coefficients other
than
R.
For this purpose we use for coefficients
R
the integral representation of the
t
Expressing polynomials
L
in equation (36) as a Kummer function
@
one can easily represent the coefficient
C
through the Lauricella function
FA
depending on
N
+
I
variables. In the case of
k
=
0
this is equivalent
to an earlier (ErdClyi 1936a) result. The possibility of expressing coefficients
C
through functions
"F::;
and
"Fi::
depending on lesser number of variables is an evident advantage
of
our
approach.
[
x
F2:o

Linearisation relations for Laguerrt! polynomials 1409
where
0;
=
wl(wI
+
w2)-',
wi
=
w2(wI
+
w2)-'.
Then, taking into account the recurrence relation for L,(x)
nL:( x)
=
(n
+
a)~z-,(x)
-
a~:f:(x)
+
x~zT:(x)
which is a consequence of two standard relations for the Laguerre polynomials (Erdilyi
1953, equations 10.12 (23) and (24)), we obtain the necessary recurrence equation
v,q;;"(w;,
w;)=(vI+211+
l)I~l'?l,","(w;,
w;)-w',I>,';y2"(w;,
w;)
+w;I:;&(w;,
U;)
as well as two similar relations for the indexes
12,
v2
and
1,
v.
Initial values for such
a system of equations are either the Appell functions
F2,
if
the recursion over only
one of
vl,
v2,
v
indexes is used, or the Gauss functions,
F:,
if the recursion over two
indexes is utilised.
6.
Expansion
of
the product
Lz;(u,x)L~~u,x)
in terms
of
L~I+~~[(U,
+
u,)xI
In
some special cases there is no need in using general formulae for the coefficients
C.
Let us consider,
for
example, an alternative formulation of the linearisation theorem
for the particular case of practical interest
N
=
2,
k=O,
a
=
a,
+
a?,
x=x,+x>
in
equation (34), that leads to an especially simple expression for the coefficients
C
and thus yields a non-trivial reduction rule for the functions
NF
in equations (37) and
(38).
Using in the Rodriguez formula for the Laguerre polynomials the differential identity
we obtain the following 'parametric' representation for
Lz
(x):
1
n.
L:(x) =ye" d"(h)A"+u e-h"
where d(h)
=
d/dh. Then
L~;(u,x)L~;(u,x)
=-
exp(-ux) d"l(Xl) d"l(h,) h?'+'lA;2*u2
1
n,
!
n2!
xexp[-(u{A,
+
u;A2)ux]
A1=A2=1
where
U
=
U1
+
uz
U;
=
UI/U
U;
=
U21
U.
(39)

1410
A
W
Niukkanen
Changing the variables
A,,
A2
to new variables
A=u{hl+u;A2
p=AI-A2
and taking into account that
AI
=
A
+
U$
A2zA-u;~
d(Ai)=u; d(A)+d(p) d(A2)
=
4
d(A) -db)
we have
G;
(UIX)
C;(
u2x)
1
--
-
(-1)*2
exp(-ux)u;n2+a2u;nl+ai(d(p)+
U{
d(A))"l(d(p)
-
U;
d(A))"2
n,!
n2!
x(p+~~-'A)''~+~~(p
-u{-IA)"~+~~
exp(-Aux) (40)
I*=I,Y=O.
Applying equation (34) from Niukkanen (1983) for the case
N
=
2, taking into account
that the Lauricella functions,
F,,,
arising in such an expansion are transformed into
the Gauss functions,
F:,
and writing down the functions
F:
through the Jacobi
polynomials P?3p'(x), we have
(41)
Using expansion (41) both for the product of operator binomials, and for the product
of non-operator binomials in equation (40), taking into account that
U{
+
U;
=
1,
making
appropriate differentiations, expressing the derivative with respect to
A
via relation
(39) and taking into account that, by virtue of
p
=
0,
the resulting double sum is
reduced to a simple one, we can assert finally that the coefficients
C
in the expansion
have the following simple form:
c
a1
.a2.e,
+e2
nl.n*,fl
(0;
U,,
u2)
U
;-n2u;
-"I
n!(n,
+
n2
-
n)!
nl!n2!
- -
(43)
provided that
ul
+
u2
=
1
and
aI
+
a2
=
a.
Note that the expansion (42) is a series of Clebsch-Gordan type with the constant
weight
a
=
a,
+
a2
of Laguerre polynomials in the right-hand side, whereas Bailey and
Howell expansions, for example, are modified series of Clebsch-Gordan type (for
particular values
of
parameters) with the weight indexes depending
on
the summation
variable
n
(see
8
1).
Using definitions (27) and (28), equation (41) can be easily reformulated for
hydrogen-like functions or their radial parts. Using expansion (42) in equations (31)
and (36) for
N
=
2, one may obtain alternative expansions over the Jacobi polynomials
for both the transition probability integral
If'
and the coefficient
C(k;
U,,
U,; U)
of
n-n2,n-n,l
LI
+n-n2,a2+n-nl)
x
p',,+,,-,
(U2
-
w%,L-n
(u2-
U11

Linearisation relations for Laguerre polynomials 1411
general type. In the cases
n2
=
0
or
a2
=
-n2
expansion (42) is equivalent to particular
types of expansions discussed in
0
3. If
nl
=
n2,
or
ai
+
n,
=
a,+
n2
or
nl
=
n2
and
al
=
a2,
then in the product of the Jacobi polynomials
on
the right-hand side of equation
(43) the first multiplier
or
the second one,
or
both of them, respectively, turn out to
be the Gegenbauer polynomials,
C^,
(ErdClyi 1953, equation 10.9 (4)). Since the
quantity Ck(0) has a form of
r
product (ErdClyi 1953, equation 10.9 (19)), this means
that in the case
n,
=
n2,
al
=
a*,
uI
=
u2,
the coefficient
C
in equation (43) has a simple
form of Pochhammer symbols product.
Equation (43) can be obtained with the help of another method which can also be
used to prove one of Koornwinder’s positivity theorems (see
Q
1)
in a more appropriate
and compact way. Using the integral representation
(8)
for
C
in equation (42),
transforming LP;(x) with the help
of
the Rodriguez formula, integrating by parts
n
times, applying the Leibnitz rule for the product derivative and expressing the deriva-
tives of the Laguerre polynomials with the help of the standard formula (ErdClyi 1953,
equation 10.12 (15)), we have
where
[ili21n]
denotes the set of conditions
il
3
0,
i2
5
0,
il
+
i2
=
n.
Provided that
u1
=
u2
=
1
and
a
=
al
+
a2
(see equation (l)), one can use the following known formula?
which is another particular case of the coefficients
C(
k,
T)
(see equations
(8)
and (20)).
In the case of non-negative integral
al
and
a2
this gives a representation of
C
in
equation (44) as the product of of an explicitly positive expression and the factor
(-
,)9+,2-,
.
In
other words, this method not only gives us a positivity proof for
Ci
in
equation
(1)
but also leads to an explicitly positive, i.e. containing only non-negative
contributions, expression for Koornwinder’s coefficients.
One can transform equation (44) to the form of equation (43) with the aid of the
following argument. If use is made of the Rodriguez formula for both the L in equation
(44), then putting
uI
+
u2
=
1 and
a
=
al
+
a2
(see the note following equation (43)),
one can verify that cancellation of both the exponential and the power factors takes
place in the integrand
of
equation (44). This makes it easy to proceed with integration
by parts which leads, eventually, to equation (43). Note that an equivalent approach
is applicable to the product of Jacobi polynomials. This gives an alternative method
of expanding the product P~~spl)(x)P~2.p2)( x) in terms of
P~I+~~”I+~~)(
x) (cf Vilenkin
1965).
Equation (42) can also be applied to give a new simple
proof
of
the second
Koornwinder positivity theorem (see equation
(2)).
Really, multiplying both sides of
equation (42) by
S;iS;2
we can carry out the summation over all integral
n,,
n2
with
the help of the generating function for Laguerre polynomials (ErdClyi 1953, equation
10.12
(17)).
Writing down the resulting exponential function and the polynomial L:(x)
t
This
formula is given by equation
7.414 (9)
in Gradstein and Ryzhik
(1971)
with some misprints.
The
correct form is given by equation
(45).

1412 A
W
Niukkanen
as
,F0
and
IFI,
respectively, and applying the operator
I
Fo[a
+
1
;
z
a/a~]~=,
(Niukkanen
1983) to both sides of the equation, one can obtain a simple generating function for
the coefficients
C
in equation (42).
In
the case
ai
=
a2
=
a
and
U,
+
u2
=
1
(see equation
(2)) such a function proves to be a product of the generating functionf,, corresponding
to
a
=
0,
and a simple function
f,
which can readily be represented by the Taylor series
in
S;lS;z
with the positive coefficients
C,.
Note that equation (43) gives an explicitly
positive expression for the coefficients
CO
of the second Koornwinder expansion
equation (2) for the important particular case
a
=
0.
By definition the quantities
CO
represent the coefficients in the Taylor expansion
of
fo.
Since the product of two Taylor series with positive coefficients is again a series
with positive coefficients, this argument should give, apparently, a simple positivity
proof for the coefficients of the second Koornwinder expansion equation (2)t.
7.
Modified series of Clebsch-Gordan type for the product
of
the Laguerre
polynomials
An
alternative type of linearisation theorem for the
Lz;(
u,x)L:;(u2x)
product arises
when the integral representation of
L:
polynomials via the Bessel function,
Ja,
is used:
which is obtained by a square substitution of the integration variable in the standard
integral representation for
Lz
(Erdtlyi 1953, equation 10.12 (21)). We transform each
of the two multipliers
L
by means of equation (46), denote the corresponding integration
variablesby
T~,
72
and introduce polar coordinates
7,
Q
in the
T~,
T~
plane
(
7,
=
T
cos
Q,
T~
=
7
sin
9).
Introducing an arbitrary parameter,
U,
we use the Bailey (1935b) linearisa-
tion theorem (see also equation 7.15
(7)
in Erdilyi (1953) and
§
7
in Niukkanen (1983))t
for the following values of parameters:
WI
=
(Ui/
COS
Q
w2=
(u~/u)”~
sin
Q
z
=
27(
ux)l’2
The quantity
y,
for the time being, is arbitrary, and this gives us an opportunity to
simplify the resulting expansion at a final stage. If the quantity
p
=
a,
+
a,+
1
-
y
assumes
an
integral value and
if
k
6
n,
+
n2+p
then according to equation (46), the
integral over variable
7
is expressed in terms of the Laguerre polynomial.
As
a result,
t
This reasoning resulted from the discussion which took place in course of communication with the referee.
The details of the approach will be published in a separate paper.
$
Hypergeometric series,
*F,
in equation
(47)
is just the Appell function,
F4,
in the notation of Niukkanen
(1983).

Linearisation relations for Laguerre polynomials
1413
we obtain
(48)
where
U
-k,
y
+
k;
ia,ia;
-
U1
cos2
cp,
U
0;
a,+
1,
a2+
1 (49)
By term-by-term integration of the series
on
the right-hand side of equation (49), one
can easily show that the function
G
is expressed as a hypergeometric series of higher
rank
1
T(a,
+
n,
+
1)T(a2+
n2+
1)
2 r(aI+az+nl+n2+2)
-_
-
-k,
y
+
k;
a1
+
n,
+
1,
a2+
n2
+
1
;
U,/
U,
a,+ a,+
n,+
n2+2;
al+l,
a,+
1
x
*
F:::
[
Expansion (48) still has a formal character since, though the quantity
p
=
aI
+
a,+
1
-
y
can be made an integer by choosing y, the inequality
k
S
n,
+
n2+
p,
under which the
right-hand side of equation (48) has meaning, is an outside condition and does not
follow
so
far from the properties of coefficients.
As
a result of the condition
ul
+ u2
=
U,
not only does the exponential term in equation (48) vanish, but we also arrive at the
desired selection rule,
k
s
n,
+
n2+
p,
for the function
G.
Indeed, if
U
=
u1
+
u2,
then
the arguments of the function
2F;Y
on
the right-hand side of equation (49) can be
written as
(u,/u)c0s2cp=x(1-y) (u2/u)sin2cp=y(l-x)
where x
=
cos2
Q,
y
=
u2/u.
This makes it possible to use the expansion (Burchnall
and Chaundy 1940, equation (54)):
for the function
'F
in equation (49). Writing down the Gauss functions,
F:,
in the

1414
A
W
Niukkanen
form of Jacobi polynomials, we obtain:
where
On
the right-hand side of equation
(50)
we have
0
G
r
S
k
and
r
S
p
if
p
=
0,
1,2,
. .
.
.
Introducing the integration variable,
t
=
cos
2p,
into equation (51), the function
Z(
r)
may be written, by analogy with equation
(IO),
as
a
scalar product of the Jacobi
polynomial (with the proper weight function) by some 'residual' expression, which
turns out to be polynomial if
p
+
n2
-
r
=
0,
1,2,
.
.
. .
This
condition will be satisfied
for any values of
n2,
if
r
G
p.
Since the condition
r
S
p
is satisfied for integral
p
by
virtue of the Pochhammer symbol
(-p),
on
the right-hand side of equation
(50),
we
obtain, by analogy with the relation (9), the necessary selection rule
k
s
n,
+
n2+
r
according to which the value
Z
and, respectively, the function
G
are non-zero. The
algebraic expression for the function
Z(r)
is given by the standard relation (Gradstein
and Ryzhik 1971, equation 7.391 (2)) (we introduce the missing multiplier
(a
+
l),,/n!
into the right-hand side of this formula)
5,
dt(
1
-
t)"(
1
+
t)"Pr9P'(
t)
3.
=
2P+"+l
r(p+l)r(o+l)
(a+l),
-n, n+a+p+l,p+l;
1
r(p+u+2)
-4
n!
a+l,p+u+2
Combining the obtained transformations and selection rules, we come to the
following result. Let
y
be chosen
so
that the quantity
p
=
a,
+
a,+
1
-
y
assumes a
nonnegative integer value. Then the following linearisation theorem takes place:
where
O~k~n,+n~+p
and
3
-k+r,k+y+r,a2+n2+1;
1
y-a,
+r,
a,+a2+n,+n2+r+2
2
(53)
where
0
c
r
=s
min(
k,
p).
In
the case
y
=
al
+
a2
+
1
only
a
single term with
r
=
0
survives
on
the right-hand
side of equation (53), and the linearisation theorem, equation
(52),
assumes
a
simpler

Linearisation relations for Laguerre polynomials
form:
1415
(aI+a2+2k+l)r((~I+~12+k+l)
‘E
mya,+k+i)
r(
a,+
n2+
l)T(a,
+
n,
+
1)
n,!n,!r(a,+a2+n,+n2+2)
k=O
-
-
-k,a1+a2+k+l,a2+n2+1;1
a,+
1,
a,
+
a,+
n,
+
n2+2
x
(n,
+
n2- k)!
F:
X[(UI
+
U2)X]kL::=nq2-c:+2k[(UI
+
u2)x-j
(54)
which is equivalent to an earlier (Carlitz 1957) result. This result was rediscovered
later in nuclear physics with the help of the group theoretical approach?. By taking
various normalisations into account, the coefficient
F:
appears as a Clebsch-Gordan
coefficient of
SU2,
if
a,
and
a2
are integers. Half-integral values of
a,
and
a2
have
been considered by Knyr
et
a1
(1976). Seven different formulae and recurrence relations
have been given by Raynal (1976), all these formulae being equivalent and valid for
arbitrary values of
a,
and
a2.
The case of arbitrary values of
a,
and
a2
has also been
considered more recently by Chacon
et
a1 (1979). It is worth noting that there are
8
symbols
F:
with two negative integers, and
20
with one negative integer as a con-
sequence of Whipple relations (Raynal 1978). Note also that such symbols as
F:
are
closely connected with the well known ‘Regge symbols’.
So equations
(52)
and (54) give a good way of comparing classical and group
theoretical methods. It seems that the latter are more elegant and efficient in particular
cases but the former have a wider field of application. Note that from the classical
point of view the difference between the integral, half-integral and arbitrary values of
a,
and
a2
seems to be artificial and, therefore, many difficulties can be avoided.
On
the other hand, we have a more general equation
(52)
that is more useful in applications
since
it
allows us a possibility of shifting the weight index in the Laguerre polynomials.
This expansion seems to be a ‘hard nut’ for the group theoretical methods.
In
some particular cases expansion
(54)
can be transformed to a simpler form.
In
the case
n,
=
n,
and
a,
=
a2
the series
F:
(1)
in equation (54) may be summed
up with the help of the Watson theorem (ErdClyi 1953, equation 4.4 (6)).
As
a result,
the coefficients in equation (54) assume a simple form of
r
products. If
a,
=
a2
and
U,
=
U,,
then, as in the case considered in
§
6, the Jacobi polynomial assumes a simple
form of
r
product. Note that these two cases give analogues of the Bailey and Howell
theorems (see
§
1)
with the difference that the Laguerre polynomial
on
the right-hand
sides of our expansions has the multiplier,
xk,
rather than
xZk.
It is worth noting also
that in the case
n2
=
0
the series
F:
(1)
in equation (54) reduces to the function
F:
(1)
which is summed up by means of the Gauss theorem. In this case expansion (54)
transforms into the ErdClyi (1936b) multiplication theorem (see also equation 6.14
(7)
in ErdClyi (1953)).
8.
Conclusions
It is shown that the product
xkL;;(uIx).
. .
L::(uNx)
is expressed as a linear combina-
tion of polynomials
L:(ux)
with coefficients
C
having a form of generalised hyper-
geometric series,
NF
(Niukkanen 1983).
In
some particular cases the coefficients
C
t
Technically, this approach
is
closely related to hyperspherical harmonics formalism (see, for example,
Raynal
1976).
The author is indebted to Professor
J
Raynal
for
useful discusion of the group theoretical topics.

1416
A
W
Niukkanen
assume an especially simple form:
for
N
=
1
they are expressed via the Clausen function,
F:
(
l/ui), and for k
=
0
and
N
=
2
either as a product of two Jacobi polynomials (for
the case of a series of Clebsch-Gordan type),
or
as a product of the Clausen function,
F:
(I),
by the Jacobi polynomial (in the case of a series of modified type).
On
the
one hand, these special forms of linearisation theorems give reduction rules for the
series
NF
of a particular type and, besides, allow us to represent the corresponding
series
NF
of a more general type as expansions in terms of simpler functions. On the
other hand, particular types of linearisation theorems are general enough for many
physical applications. In particular, these linearisation relations can be easily reformu-
lated for the hydrogen-like functions, which considerably facilitates the analytical
formulation of the multicentre integral problem in variational calculations of molecular
electron wavefunctions.
Many types of expansions known from literature turn out to be particular cases of
expansions presented in this paper that result in unifying numerous relations involving
Laguerre polynomials.
Acknowledgments
I
express my gratitude to Professor L
A
Gribov whose kind attention to the present
work greatly contributed to its development and completion.
I am very indebted to Professor
J
Raynal for useful discussion of the
SU2
formalism.
I
highly appreciate the referee’s suggestions which inspired some new proofs and
relations.
I
am very obliged to Professor
N
Ya Vilenkin for general discussion of the
results presented in this paper.
Note added in prooj
Useful additional coniments on
operatorfactorisation relations,
like that mentioned at
the end
of
I6 in connection with positivity proofs, are given in Niukkanen (1984b).
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