General linearization formulae for products of continuous hypergeometric-type polynomials

Abstract

The linearization of products of wavefunctions of exactly solvable potentials often reduces to the generalized linearization problem for hypergeometric polynomials (HPs) of a continuous variable, which consists of the expansion of the product of two arbitrary HPs in series of an orthogonal HP set. Here, this problem is algebraically solved directly in terms of the coefficients of the second-order differential equations satisfied by the involved polynomials. General expressions for the expansion coefficients are given in integral form, and they are applied to derive the connection formulae relating the three classical families of hypergeometric polynomials orthogonal on the real axis (Hermite, Laguerre and Jacobi), as well as several generalized linearization formulae involving these families. The connection and linearization coefficients are generally expressed as finite sums of terminating hypergeometric functions, which often reduce to a single function of the same type; when possible, these functions are evaluated in closed form. In some cases, sign properties of the coefficients such as positivity or non-negativity conditions are derived as a by-product from their resulting explicit representations.

Full text

General linearization formulae for products of continuous
hypergeometric-type polynomials
JS
´
anchez-Ruiz†‡,PLArt
´
es§, A Mart´
ınez-Finkelshtein‡§ and J S Dehesa‡k
† Departamento de Matem´
aticas, Universidad Carlos III de Madrid, Avda. de la Universidad 30,
28911 Legan´
es, Madrid, Spain
‡ Instituto Carlos I de F´
ısica Te´
orica y Computacional, Universidad de Granada, 18071 Granada,
Spain
§ Departamento de Estad´
ıstica y Matem´
atica Aplicada, Universidad de Almer´
ıa, La Ca˜
nada,
04120 Almer´
ıa, Spain
kDepartamento de F´
ısica Moderna, Universidad de Granada, 18071 Granada, Spain
Abstract. The linearization of products of wavefunctions of exactly solvable potentials often
reduces to the generalized linearization problem for hypergeometric polynomials (HPs) of a
continuous variable, which consists of the expansion of the product of two arbitrary HPs in
series of an orthogonal HP set. Here, this problem is algebraically solved directly in terms of
the coefficients of the second-order differential equations satisfied by the involved polynomials.
General expressions for the expansioncoefficients are given in integral form, and they are applied to
derive the connection formulae relating the three classical families of hypergeometric polynomials
orthogonal on the real axis (Hermite, Laguerre and Jacobi), as well as several generalized
linearization formulae involving these families. The connection and linearization coefficients are
generally expressed as finite sums of terminating hypergeometric functions, which often reduce to
a single function of the same type; when possible, these functions are evaluated in closed form. In
some cases, sign properties of the coefficients such as positivity or non-negativity conditions are
derived as a by-product from their resulting explicit representations.
1. Introduction
Consider the second-order differential operator
F[y](x) =σ(x)y00(x ) +τ(x)y0(x) (1)
where σ(x) and τ(x) are polynomials whose degrees are not greater than two and one,
respectively. If |τ0|+|σ00|6=0, then for every n∈N,Fhas a polynomial eigenfunction
y(x) =yn(x) of degree ncorresponding to the eigenvalue
λn=nτ 0+1
2n(n −1)σ 00.
These polynomials are usually called hypergeometric-type polynomials (or continuous
hypergeometric-type polynomials, in contrast with those which appear as eigenfunctions of
second-order linear difference operators); by means of linear changes of the variable, they
can be reduced to one of the four classical families (Hermite, Laguerre, Jacobi and Bessel).
The hypergeometric-type polynomials are involved in the classical eigenfunctions of singular
Sturm–Liouville equations [1–6] as well as in the quantum mechanical wavefunctions of the
exactly solvable potentials [5,7–10], and they are used in the mathematical modelling of a
great amount of physical and chemical phenomena [3,5,11–14].
1

This paper deals with the general hypergeometric linearization problem, which is the
problemof findingthe coefficientsgnmk inthe expansionof theproduct oftwo(hypergeometric)
polynomials, pn(x)qm(x ), in terms of an arbitrary sequence of orthogonal hypergeometric
polynomials {yk(x)},
pn(x)qm(x ) =
n+m
X
k=0
gnmkyk(x). (2)
Note that, in this setting, the polynomials pn,qmand ykmay belong to three different
hypergeometric families. Particular cases of this problem have been matter of intensive
research (see, e.g., [12, 15] and the bibliography therein), and frequently receive different
names. For example, when the polynomials pn,qmand ykare solutions of the same differential
equation (1), this is usually called the (standard) linearization [12] or Clebsch–Gordan-type
problem [16] for hypergeometric polynomials (the name Clebsch–Gordan is attached because
the structure is similar to the Clebsch–Gordan series for spherical functions [17]). On the other
hand, taking qm(x) =1 in (2), we are faced with the so-called connection problem, which for
pn(x) =xnis known as the inversion problem for the family {yk(x)}.
The literature on the standard linearization and connection problems is extremely vast,
and a variety of methods and approaches for computing the coefficients gnmk in (2) have been
developed. For the classical families of polynomials, explicit expressions have been obtained,
usually in terms of generalized hypergeometric series, exploiting with this purpose several
of their characterizing properties: recurrence relations, generating functions, orthogonality
weights,etc (see,e.g., [3,12,16,18–20]). Another,rathergeneral, approach allows computation
of the standard linearization and connection coefficients recursively (see, e.g., [15,20, 21]).
In contrast, the general linearization problem has not yet been solved [12, 15], although
some partial results are known for Jacobi polynomials [14], and explicit expressions for the
coefficients gnmk when pn,qmand ykare Laguerre polynomials with different parameters have
also been given [16]. This is somewhat surprising, because there are numerous fundamental
and applied questions related to this problem, such as the linearization of products of basis-set
functions associated to shape-invariant potentials [10], the transformation formulae between
quantum mechanical wavefunctions in different coordinate systems (e.g. the bound-state
wavefunctionsonL2(R3)inspherical andparabolic coordinates[22]), theinterbasisexpansions
for potentials of equal [23] and different [24] dimensionality (e.g. the passage formulae from
the R3hydrogen wavefunctions to R4oscillator wavefunctions [22]), the determination of
the Talmi–Brody–Moshinsky coefficients [25] so widely used in nuclear structure, and the
evaluation of two-centre, two- and three-electron integrals in variational atomic analysis [13].
Themain aimof thispaper isto producesome analytic(in general,integral) representations
forthe coefficientsgnmk in (2), extending the method proposed very recentlyby the authors [26]
to solve the inversion, connection, and standard linearization problems for hypergeometric
polynomials. These analytic expressions of gnmk are given directly in terms of the coefficients
of the differential equations corresponding to the hypergeometric polynomials pn,qnand yn,
which is often a desirable feature. In fact, products of hypergeometric polynomials appear
naturally when dealing with eigenfunction systems of Sturm–Liouville equations [27,28] and
the Schr¨
odinger equations associated to central potentials [7,17]. Furthermore, the quantum
mechanical equation of motion of numerous physical systems with a shape-invariant potential
can be reduced to a second-order differential equation of hypergeometric character [5,7,8].
In many applications of orthogonal polynomials, it is often important to know whether the
linearization and connection coefficients are positive or non-negative (see, e.g., [12,29,30]).
During the last decades, several sufficient conditions for thesesign properties to hold have been
derived, both for general polynomials and for the classical families (see, e.g., [12,30–34]).
2

Although the focus of the present paper is on the explicit computation of the coefficients,
rather than on the study of their sign properties, we shall find that in many cases these sign
properties become obvious from the representations given here; typically, this happens when
the coefficients can be written as a sum of terms which are shown to have the same sign by
taking into account the comments after equation (A.1).
The paper is structured as follows. First, in section 2, the general expressions for the
linearization coefficients are obtained, and the particular cases of the standard linearization
and connection problems are singled out. We also find the remarkable result that, for
hypergeometric polynomials, the linearization coefficients can always be written as a finite
combinationof connectioncoefficients. Later,for illustration, theconnection formulaerelating
thethree classicalfamilies ofhypergeometric polynomialsorthogonal on the real line(Hermite,
Laguerre and Jacobi) are obtained (section 3). The linearization problems for Hermite and
Laguerrepolynomials arerevisited, and closed expressions arealso obtainedfor thecoefficients
of the Hermite expansion of products involving Hermite polynomials (section 4). Some
concluding remarks are given in section 5, and, finally, the definitions and identities for special
functions used throughout the paper are collected in the appendix.
2. General results
We begin by introducing some notation. Below Pis the class of all polynomials with real
coefficients. If {yn(x)}n∈Nstands for the sequence of monic hypergeometric polynomials
corresponding to operator (1), then we denote by yn,k (x), with n, k ∈N, the monic polynomial
eigenfunction of degree nof the operator
Fk[y](x) =σ(x)y00(x ) +τk(x)y 0(x) τk(x ) =τ(x)+kσ 0(x) (3)
with |τ0
k|+|σ00|6=0, so that
yn(x) =yn,0(x ) y0
n,k (x) =ny n−1,k+1(x). (4)
Analogous notation will be used for other sequences of monic hypergeometric polynomials as
well. For the sake of brevity, below we omit the subindex kwhen k=0.
An explicit expression for these polynomials is provided by the Rodrigues formula. Fix
ω(x) =exp Zxτ(t)−σ0(t )
σ(t) dtω
k(x) =[σ(x)]kω(x) k >0.(5)
Then ωk(x) is a solution of the so-called Pearson’s equation, [σ(x)ω
k(x)]0=τk(x )ωk(x).If
A
n,k =
n−1
Y
j=0τ0
k+n+j−1
2σ00−1
=
n−1
Y
j=0τ0+n+j+2k−1
2σ00−1(6)
then
yn,k (x) =An,k
ωk(x)
dnωn+k(x)
dxn.(7)
In what follows, we assume, additionally, the existence of two values −∞ 6a<b6∞
such that
ω(x) ∈C(a,b) lim
x→a+ω1(x)x k=lim
x→b−ω1(x)x k=0k>0.(8)
This assumption leads to orthogonality of {yn(x)}with respect to the weight function ω(x) on
the interval [a, b] (see [5]),
Zb
a
yn(x)ym(x )ω(x) dx=~nδn,m ~n=(−1)nn!Anγn(9)
3

where {γn}denotes the sequence of generalized moments of ω(x),
γn=Zb
a
ωn(x) dx. (10)
Likewise,everysequence {yn,k (x)}is thenorthogonal with respectto the weight function ωk(x)
on the same interval,
Zb
a
yn,k(x)ym,k (x)ωk(x ) dx=~n,kδn,m ~n,k =(−1)nn!An,k γn+k.(11)
Let us denote by {pn(x)}and {qm(x )}two (possibly different) polynomial sequences,
not necessarily orthogonal or hypergeometric. The orthogonality relation (9) for the sequence
{yn(x)}leads to thefollowing explicitexpression for the Fouriercoefficients gnmk of the general
linearization formula (2):
gnmk =1
~kZb
a
pn(x)qm(x )yk(x)ω (x) dx. (12)
This equation enables us to compute gnmk in a straightforward way provided that the explicit
expressions of the three polynomials involved are known. However, as we shall show in the
following, the hypergeometric nature of the polynomials can be used to derive alternative
expressions for the linearization coefficients, which may be written in terms of the coefficients
of the underlying differential operators and are computationally more efficient.
Taking advantage of equations (7) and (9), we can rewrite (12) as
gnmk =(−1)k
k!γkZb
a
pn(x)qm(x ) dkωk(x)
dxkdx. (13)
It is easy to verify that for every P∈Pthere exists a Q∈Psuch that
dj
dxj[ωk(x)P (x )]=ωk−j(x)Q(x) j 6k.
Thus,integrating by partsktimes andtaking intoaccountthe boundaryconditions (8),Leibniz’s
rule allows us to rewrite (13) in the form
gnmk =1
k!γk
j+
X
j=j−k
jZb
a
djpn(x)
dxj
dk−jqm(x)
dxk−jωk(x) dx(14)
where j−=max(0,k−m),j+=min(k, n).
In comparison with (12), (14) has two main advantages. Firstly, it does not require the
knowledge of the explicit expression of the polynomials yk(x). Secondly, although in both
cases gnmk is expressed as a three-level summation of terms which are essentially the moments
of a weight function, the degree reduction by derivatives provides a smaller number of terms
when (14) is used. For instance, the number of terms in the expressions for gnnn given by (12)
and (14) are, respectively, (n +1)
3and
n
X
j=0
(n −j+1)(j +1)=1
6(n +1)(n +2)(n +3).
Equation (14) is especially useful when the derivatives of the polynomials pn(x) and qm(x )
have simple expressions. In particular, taking pn(x) =xnand qm(x) =1, we can obtain a
solution for the inversion problem in terms of the moments of the weights ωk(x) [26]: the
coefficient ιnk in the expansion
xn=
n
X
k=0
ιnkyk(x)
4

is given by
ιnk =n
k1
γkZb
a
xn−kωk(x) dx.
Let us assume now that both pn(x) and qm(x) are also monic polynomials of
hypergeometric type. Formula (14) can then be written, using (4), in the form
gnmk =1
γk
j+
X
j=j−n
j m
k−jInmk(j) Inmk (j ) =Zb
a
pn−j,j(x)qm+j−k,k−j(x)ωk(x) dx.
(15)
This equation is feasible for the computation of the generalized linearization coefficients
whenever the explicit expressions of the polynomials involved in the integral are known,
as is the case, e.g., for the classical hypergeometric families (see the appendix). For general
families of polynomials, when only the coefficients of the corresponding differential operators
are available, we can make one more step and find an equivalent expression for gnmk that
does not require the knowledge of the explicit expressions of the polynomials. Initially, we
restrict our attention to the standard linearization and connection problems, for which simpler
formulae can be derived from (15).
In the particular case when the three families of hypergeometric polynomials coincide,
we have a solution for the (standard) linearization (or Clebsch–Gordan) problem. Using the
orthogonality property (9), equation (2) simplifies to
yn(x)ym(x ) =
n+m
X
k=|n−m|
lnmkyk(x) (16)
and (15) now reads as
lnmk =1
γk
j+
X
j=j−n
j m
k−jInmk(j) Inmk (j ) =Zb
a
yn−j,j(x)ym+j−k,k−j(x)ωk(x) dx.
(17)
Using equation (7) for yn−j,j(x), the previous expression for the integrals Inmk(j ) can be
written as
Inmk(j) =An−j,j Zb
a
dn−jωn(x)
dxn−jym+j−k,k−j(x)[σ(x)]k−jdx.
Observe that by (4) and (7), for 0 6j6n,
dj
dxj1
ωk(x)
dnωn+k(x)
dxn=n!
(n −j)!
An−j,k+j
An,k
1
ωk+j(x)
dn−jωn+k(x)
dxn−j.
Thus, integrating by parts n−jtimes and taking into account the boundary conditions (8),
Inmk(j) =(−1)n−jAn−j,j Zb
a
ωn(x) dn−j
dxn−j(ym+j−k,k−j(x)[σ(x)]k−j)dx.
Using the Leibniz rule and equation (4),
Inmk(j) =(−1)n−jAn−j,j
i+
X
i=i−n−j
i(m −k+j)!
(m −n−k+2j+i)!
×Zb
a
ym−n+2j−k+i,k−2j+n−i(x) di[σ(x)]k−j
dxiω
n(x) dx
5

wherei−=max(0,n−m+k−2j),i+=min{n−j, (k−j)deg[σ(x)]}. Now wecan substitute
the explicit expression of the remaining polynomial, given by the Rodrigues formula (7), and
integrate by parts again, which yields
Inmk(j) =(−1)m+k+jAn−j,j
i+
X
i=i−n−j
i(−1)i(m −k+j)!
(m −n−k+2j+i)!Am−n+2j+i−k,k−2j+n−i
×Zb
a
ωm(x) dm−n+2j−k+i
dxm−n+2j−k+i[σ(x)]2j−k+idi[σ(x)]k−j
dxidx. (18)
In spite of its apparent complexity, this formula has the advantage that no derivatives of
the weight functions are involved; it does not make use of the explicit expressions of the
polynomials either. In fact, if we know σ(x)we can easily express the integrals appearing in
(18) as a linear combination of the moments of the weight function ωm(x), which makes this
equation suitable for symbolic manipulation.
Let us consider now the connection problem
pn(x) =
n
X
k=0
cnkyk(x) (19)
where {pn(x)}is the sequence of monic polynomial eigenfunctions of the operator
G[y](x) =˜σ(x)y00(x) +˜τ(x)y0(x) (20)
and deg[pn(x)]=n. Taking into account that q0(x) =1 for any sequence {qn(x)}of monic
polynomials, we readily see that cnk =gn0k, so that the connection coefficients cnk can be
obtained as the particular case m=0 of both (12) and (15). Again, it turns out to be much
more convenient to use (15), which leads to
cnk =1
γkn
kInk Ink =Zb
a
pn−k,k(x)ωk(x) dx. (21)
Using this formula, cnk is expressed as a simple summation with n−k+ 1 terms, while (12)
would give a double summation with (n +1)(k +1)terms. Furthermore, (21) does not require
the use of the explicit expression of yk(x).
By equation (7), the previous formula for Ink can be written as
Ink =˜
An−k,k Zb
a
ωk(x)
˜ωk(x)
dn−k˜ωn(x)
dxn−kdx(22)
or, equivalently, integrating by parts n−ktimes,
Ink =(−1)n−k˜
An−k,k Zb
a˜ωn(x) dn−k
dxn−kωk(x)
˜ωk(x) dx. (23)
A common situation in connecting polynomials of the same family, but with different
parameters, is when ˜σ(x) =σ(x). In this case, if we put ˜ω(x) =f (x)ω(x), the previous
equation takes the form
Ink =(−1)n−k˜
An−k,k Zb
a
f(x)ω
n(x) dn−k
dxn−k1
f(x)dx(24)
which may be useful if the derivatives of 1/f (x) have simple expressions.
For arbitrary families of polynomials, the generalized linearization problem can always
be reduced to a combination of two connection and one standard linearization problems: if we
6

write
pn(x) =
n
X
s=0
cns (p)ys(x) qm(x) =
m
X
t=0
cmt (q)yt(x )
ys(x)yt(x ) =
s+t
X
k=|s−t|
lstkyk(x)
(25)
then we have
pn(x)qm(x ) =
n
X
s=0
m
X
t=0
s+t
X
k=|s−t|
(cns (p)cmt (q)lstk)yk(x)
so that the generalized linearization coefficients gnmk in (2) may be computed as
gnmk =X
|s−t|6k6s+t
cns (p)cmt (q)lstk.(26)
It is a remarkable fact that, in the case when all the involved polynomials are of hypergeometric
type, equation (15) enables us to express the linearization coefficients in terms only of two
connection coefficients, namely those corresponding to the expansions of the polynomials
pn−j,j(x) and qm+j−k,k−j(x) in series of the {yr,k (x)},
pn−j,j(x) =
n−j
X
r=0
c(j,k)
n−j,r(p)yr, k (x) qm+j−k,k−j(x) =
m+j−k
X
s=0
c(k−j,k)
m+j−k,s(q)ys,k(x). (27)
Substituting these expressions into (15) and using the orthogonality relation (11), we obtain
gnmk =1
γk
j+
X
j=j−n
j m
k−jInmk(j)
Inmk(j) =
r+
X
r=0
c(j,k)
n−j,r(p)c(k−j,k)
m+j−k,r(q)~r,k
(28)
where r+=min(n −j, m +j−k). In particular, for the standard linearization coefficients
lnmk given by (17) the previous formula does apply with p=q=y, and we can omit the
arguments of the connection coefficients to simplify the notation.
Next, we shall illustrate the application of the general results given in this section by using
them to find explicit expressions for the coefficients in the connection formulae relating the
three classical families of monic polynomials, orthogonal on the real axis (Hermite, Laguerre
and Jacobi), as well as in several linearization formulae involving these families. In fact,
as a rule we give two different expressions for each set of coefficients: namely, as a sum of
binomial type and asa hypergeometric series(see equation (A.2)); theconversion ofthe former
into the latter has been either carried out or checked using the Mathematica package HYP by
C Krattenthaler [35]. All the necessary definitions and well known identities are gathered in
appendix A.1, which we shall often make use of without explicit reference to it.
3. Connection formulae
Leaving aside the trivial case of the Hermite–Hermite connection, for which we simply have
cnk =δn,k, there are eight connection formulae of the form (19) relating the Hermite, Laguerre
and Jacobi families of orthogonal polynomials. In this section, we obtain explicit expressions
for the connection coefficients of these eight problems using equations (21)–(24). Most of
these expressions are already known in the literature, while others seem to be new; in the
former case, references where the results can be found are indicated. Likewise, whenever we
find sign properties for the coefficients that are already known, the corresponding references
are also indicated.
7

3.1. Expansions in series of Hermite polynomials
Following from (21) and (22),
pn(x) =
n
X
k=0
cnkHk(x) cnk =1
√πn
kInk (29)
where
Ink =Z∞
−∞
pn−k,k(x)e−x2dx=˜
An−k,k Z∞
−∞
dn−k˜ωn(x)
dxn−k
e−x2
˜ωk(x) dx. (30)
3.1.1. Connection with Laguerre polynomials. For pn(x) =L(α)
n(x), the second expression
for Ink in (30) reads
Ink =(−1)n−kZ∞
−∞
dn−kxα+ne−x
dxn−k
e−x2
xα+ke−xdx.
Using Leibniz’s rule to evaluate the derivative in the right-hand side, together with the well
known moments of the weight exp(−x2), we find that
Ink =(−1)n−k0(n +α+1)
[(n−k)/2]
X
j=0n−k
2j0(j +1
2)
0(k +2j+α+1)
where, as usual, the square brackets denote integer part of the expression within. The same
result can be obtained more easily by substituting in the first formula of (30) the explicit
expression of pn−k,k(x) =L(α +k)
n−k(x) given in (A.11). Thus, with account of (29), we obtain
(cf [18, p 216])
cnk =(−1)n−kn
k[(n−k)/2]
X
j=0n−k
2j1
2j
(k +2j+α+1)
n−k−2j
=(−1)
n−k(k +α+1)
n−kn
k
2F
2k−n
2,k−n+1
2
k+α+1
2,k+α
2+1



1
4.(31)
From the first of these expressions, we readily see that the sign of cnk is (−1)n−k.
3.1.2. Connection with Jacobi polynomials. For pn(x) =P(α,β )
n(x), application of Leibniz’s
rule to the second expression for Ink in (30) gives
Ink =(−1)n−k
(n +k+α+β+1)
n−k
n−k
X
j=0
(n −j+α+1)
j(k +j+β+1)
n−k−j
×(−1)
jZ∞
−∞
Bn−k,j(x )e−x2dx
where Bn−k,j(x ) =n−k
j(1−x)n−k−j(1+x)jis a Bernstein polynomial. Taking advantage
of the properties of Bn,k (x), it is easy to expand the last integral as a linear combination of the
momentsof the weight exp(−x2)or to compute it recursively. However, a simpler resultcan be
obtained using in the first formula of (30) the explicit expression of pn−k,k(x) =P(α+k,β+k)
n−k(x)
given by (A.12), which leads to
Ink =
n−k
X
j=0n−k
j2n−k−j(k +j+α+1)
n−k−j
(n +k+j+α+β+1)
n−k−jZ∞
−∞
(x −1)je−x2dx.
8

Evaluatingtheintegralsinthe right-handside, theexpressionforcnk followsin astraightforward
way from (29):
cnk =n
kn−k
X
j=0n−k
j(−1)j2n−k−j(k +j+α+1)
n−k−j
(n +k+j+α+β+1)
n−k−j
[j/2]
X
m=0j
2m1
2m
.(32)
The sum over mcan be written as a 2F0hypergeometric function of unit argument and upper
parameters −j/2, (1−j)/2. Alternatively, interchanging the order of summation and shifting
the index jto l=j−2m, the previous formula reads
cnk =n
k[(n−k)/2]
X
m=0n−k
2m1
2m
×
n−k−2m
X
l=0n−k−2m
l(−1)
l2n−k−2m−l(k +l+2m+α+1)
n−k−2m−l
(n +k+l+2m+α+β+1)
n−k−2m−l
=n
k
[(n−k)/2]
X
m=0n−k
2m2n−k−2m(1
2)
m(k +2m+α+1)
n−k−2m
(n +k+2m+α+β+1)
n−k−2m
×2F
12m−n+k, n +k+2m+α+β+1
k+2m+α+1 



1
2.(33)
In the Gegenbauer case, when α=β, (A.6) leads to a closed expression for cnk,
cnk =n
k[(n−k)/2]
X
m=0n−k
2m0(m +1
2)0( n+k+1
2+m+α)
0(n +α+1
2)0( k−n+1
2+m)
=n
k0(1
2)0( n+k+1
2+α)
0(n +α+1
2)0( k−n+1
2)2F0k−n
2,n+k+1
2+α
−


1.
We readily see that cnk vanishes whenever n−kis odd. Therefore, writing n−k=2rwith r
integer, the connection formula simplifies to [18, p 284]
P(α,α)
n(x) =
[n/2]
X
r=0
cnr Hn−2r(x)
cnr =n
2rr
X
m=02r
2m(1
2)m(1
2)r−m
(1
2−n−α)r−m
=n
2r(1
2)r
(1
2−n−α)r
2F0−r, n −r+α+1
2
−1.
(34)
3.2. Expansions in series of Laguerre polynomials
Following from (21) and (22),
pn(x) =
n
X
k=0
cnkL(α)
k(x) cnk =1
0(k +α+1)n
k
I
nk (35)
where
Ink =Z∞
0pn−k,k(x) xα+ke−xdx=˜
An−k,k Z∞
0
dn−k˜ωn(x)
dxn−k
xα+ke−x
˜ωk(x) dx. (36)
9

3.2.1. Connection with Hermite polynomials. For pn(x) =Hn(x), the first expression for
Ink in (36) reads
Ink =Z∞
0Hn−k(x)x α+ke−xdx.
Thus, by (A.10),
Ink =
[(n−k)/2]
X
j=0n−k
2j(−1
4)j(2j)!0(n −2j+α+1)
j!
and (35) then yields [18, p 207]
cnk =n
k[(n−k)/2]
X
j=0n−k
2j(−1)j1
2j
(k +α+1)
n−k−2j
=(k +α+1)
n−kn
k
2F
2k−n
2,k−n+1
2
−n−α
2,−n−α+1
2


−1
4.(37)
3.2.2. Connection with Laguerre polynomials of different parameters. For pn(x) =L(β )
n(x),
the second expression for Ink in (36) gives
Ink =(−1)n−kZ∞
0
dn−k(xβ+ne−x)
dxn−kxα−βdx.
Integrating by parts n−ktimes (or, equivalently, by direct use of equations (23) or (24)), we
readily find that
Ink =(−1)n−k0(k +α+1)(β −α)n−k
and, with account of (35), one has (cf [1, p 192], [18, p 209]),
cnk =n
k(−1)n−k(β −α)n−k.(38)
From this result, we readily see that the sign of cnk is (−1)n−kif β>α[30,31, 33], while
cnk >0ifβ−αis a negative integer. Finally, if β−α<0 is not an integer, cnk is non-negative
provided that α−β>n−k−1, so that all the connection coefficients are non-negative if
α−β>n−1.
3.2.3. Connection with Jacobi polynomials. For pn(x) =P(γ ,δ)
n(x), the first equation in (36)
gives
Ink =Z∞
0P(γ +k,δ+k)
n−k(x)xα+ke−xdx.
Using the explicit expression of the monic Jacobi polynomials (A.12), we find that
Ink =
n−k
X
j=0n−k
j2n−k−j(k +j+γ+1)
n−k−j
(n +k+j+γ+δ+1)
n−k−jZ∞
0(x −1)jxα+ke−xdx.
Evaluatingtheintegralsinthe right-handside, theexpressionforcnk followsin astraightforward
way from (35):
cnk =n
kn−k
X
j=0n−k
j2n−k−j(k +j+γ+1)
n−k−j
(n +k+j+γ+δ+1)
n−k−j
j
X
m=0j
m
(−1)
j−m
(k +α+1)
m
.(39)
10

The sum over mcan be written as a 2F0hypergeometric function of unit argument and upper
parameters −j,k+α+ 1. Interchanging the order of summation and shifting the index jto
l=j−m, we obtain the alternative expression
cnk =n
kn−k
X
m=0
(k +α+1)
m
n−k−m
X
l=0n−k
m+lm+l
m
×(−1)l2n−k−m−l(k +m+l+γ+1)
n−k−m−l
(n +k+m+l+γ+δ+1)
n−k−m−l
=n
k
n−k
X
m=0n−k
m2
n−k−m
(k +α+1)
m
(k +m+γ+1)
n−k−m
(n +k+m+γ+δ+1)
n−k−m
×2F
1m+k−n, n +k+m+γ+δ+1
k+m+γ+1 



1
2.(40)
In the Gegenbauer case, when γ=δ, equation (A.6) leads to
cnk =n
k01
2
0(n +γ+1
2)
n−k
X
m=0n−k
m(k +α+1)
m
0(m+n+k+1
2+γ)
0(m+k−n+1
2).
We readily see that the mth term in this summation vanishes whenever n−k−mis odd.
Therefore, writing n−k−m=2rwith rinteger, the expression of the connection coefficients
simplifies to
cnk =n
k[(n−k)/2]
X
r=0n−k
2r(−1)r(1
2)r(k +α+1)
n−k−2r
(n −r+γ+1
2)r
=(k +α+1)
n−kn
k
2F
3k−n
2,k−n+1
2
−n−α
2,−n−α+1
2,−n−γ+1
2



1
4.(41)
The particular case γ=0 of this formula, which corresponds to the expansion of Legendre
polynomials in series of Laguerre polynomials, is given in [18, p 208].
3.3. Expansions in series of Jacobi polynomials
Following from (21) and (22),
pn(x) =
n
X
k=0
cnkP(α,β)
k(x)
cnk =0(2k+α+β+2)
2
2k+α+β+10(k +α+1)0(k +β+1)n
k
I
nk
(42)
where
Ink =Z1
−1pn−k,k(x)(1−x)k+α(1+x)k+βdx
=˜
An−k,k Z1
−1
dn−k˜ωn(x)
dxn−k
(1−x)k+α(1+x)k+β
˜ωk(x) dx. (43)
3.3.1. Connection with Hermite polynomials. For pn(x ) =Hn(x), the first equation in (43)
gives
Ink =Z1
−1Hn−k(x)(1−x)k+α(1+x)k+βdx.
11

Substituting the explicit expression of the monic Hermite polynomials (A.10), and using (A.7),
Ink =(−1)n−k22k+α+β+10(k +α+1)0(k +β+1)
0(2k+α+β+2)
×
[(n−k)/2]
X
m=0n−k
2m(−1
4)m(2m)!
m!2F1k−n+2m, k +β+1
2k+α+β+2 


2

so that from equation (42) we obtain
cnk =(−1)n−kn
k[(n−k)/2]
X
m=0n−k
2m(−1)m1
2m
2F1k−n+2m, k +β+1
2k+α+β+2 


2
.(44)
In the particular case of Gegenbauer polynomials, when α=β, (A.5) implies that cnk
vanishes whenever n−kis odd. Then, writing n−k=2rwith rinteger, the connection
formula simplifies to
Hn(x) =
[n/2]
X
r=0
cnr P(α,α)
n−2r(x)
cnr =n
2rr
X
m=02r
2m(−1)m(1
2)m(1
2)r−m
(n −2r+α+3
2)r−m
=n
2r(1
2)r
(n −2r+α+3
2)r
2F0−r, r −n−α−1
2
−−1.
(45)
An equivalent form of this expression is given in [18, p 284].
3.3.2. Connection with Laguerre polynomials. For pn(x) =L(γ )
n(x), by the first equation in
(43) we have
Ink =Z1
−1L(γ +k)
n−k(x)(1−x)k+α(1+x)k+βdx.
Then, by (A.7) and (A.11),
Ink =(−1)n−k22k+α+β+1 0(k +α+1)0(k +β+1)
0(2k+α+β+2)
×
n−k
X
m=0n−k
m(k +m+γ+1)
n−k−m2F
1−m, k +β+1
2k+α+β+2


2
.
Finally, using (42), we find that
cnk =(−1)n−kn
kn−k
X
m=0n−k
m(k +m+γ+1)
n−k−m2F
1−m, k +β+1
2k+α+β+2


2
.(46)
Again, for Gegenbauer polynomials (α=β), equation (A.5) leads to
cnk =(−1)n−kn
k[(n−k)/2]
X
m=0n−k
2m(1
2)m(k +2m+γ+1)
n−k−2m
(k +α+3
2)m
=(−1)n−k(k +γ+1)
n−kn
k
2F
3k−n
2,k−n+1
2
k+α+3
2,k+γ+1
2,k+γ
2+1



1
4(47)
and the sign of these coefficients is readily shown to be (−1)n−k. The particular case α=0
of (47), which corresponds to the expansion of Laguerre polynomials in series of Legendre
polynomials, is given in [18, p 216].
12

3.3.3. Connection with Jacobi polynomials of different parameters. For pn(x) =P(γ ,δ)
n(x),
the first equation in (43) yields
Ink =Z1
−1P(γ +k,δ+k)
n−k(x)(1−x)k+α(1+x)k+βdx.
Using the explicit expression of the monic Jacobi polynomials (A.12), together with (A.7), we
find that
Ink =2n+k+α+β+1 0(k +β+1)
×
n−k
X
m=0n−k
m(−1)
m
(k +m+γ+1)
n−k−m
0(k +m+α+1)
(n +k+m+γ+δ+1)
n−k−m
0(2k+m+α+β+2).
Then, from (42), we obtain the connection formula
cnk =n
kn−k
X
m=0n−k
m(−1)m2n−k(k +α+1)
m
(k +m+γ+1)
n−k−m
(2k+α+β+2)
m
(n +k+m+γ+δ+1)
n−k−m
=2
n−k(k +γ+1)
n−k
(n +k+γ+δ+1)
n−kn
k

×3F
2k−n, n +k+γ+δ+1,k+α+1
k+γ+1,2k+α+β+2 


1
(48)
which, according to Askey [12, 29], was first derived by Feldheim [36]. A complete
discussion of the non-negativity cases of these coefficients can be found in [32] (see also
[29, 30, 33, 34]). As pointed out by Askey [12], there are three important particular cases
when the hypergeometric function in (48) can be evaluated in closed form by use of standard
summation formulae.
For Gegenbauer polynomials, when α=βand γ=δ, the classical Watson summation
theorem (A.8) leads to
cnk =n
k0(k +α+3
2)0( n+k+1
2+γ)0(1
2)0(α −γ+1)
0(n +γ+1
2)0( n+k+3
2+α)0(k−n+1
2)0( k−n
2+α−γ+1).
We readily see that cnk vanishes whenever n−kis odd. Therefore, writing n−k=2rwith r
integer, the connection formula simplifies to
P(γ,γ )
n(x) =
[n/2]
X
r=0
cnr P(α,α)
n−2r(x)
cnr =n
2r(1
2)r(γ −α)r
(n −r+γ+1
2)r(n −2r+α+3
2)r
.
(49)
Now we find that cnr >0ifγ>α[12], while the sign of cnr is (−1)rif α−γ>r−1, so
that all the connection coefficients have sign (−1)rif α−γ>[n/2] −1.
On the other hand, in the particular case when α=γ, the 3F2hypergeometric function
in (48) reduces to a 2F1function of unit argument, which can be evaluated in closed form by
means of the Chu–Vandermonde theorem (A.4). We thus obtain,
cnk =n
k2n−k(k +α+1)
n−k(k −n+β−δ+1)
n−k
(n +k+α+δ+1)
n−k(2k+α+β+2)
n−k
.(50)
These coefficients have sign (−1)n−kif δ>β, while they are non-negative if β−δis a positive
integer [12]. If β−δ>0 is not an integer, then the connection coefficients are non-negative
provided that β−δ>n−k−1, so that all of them are non-negative if β−δ>n−1.
13

A similar simplification of (48) can be achieved when β=δby means of the Pfaff–
Saalsch¨
utz formula (A.9), which leads to
cnk =n
k(−2)n−k(k +β+1)
n−k(k −n+α−γ+1)
n−k
(n +k+β+γ+1)
n−k(2k+α+β+2)
n−k
.(51)
In fact, this case turns out to be equivalent to the previous one because of the symmetry relation
P(α,β)
n(−x)=(−1)nP(β,α)
n(x), and thesame happens forthe signof the connectioncoefficients.
Now, they are positive if γ>α[3,12,31, 33], while their sign is (−1)n−kif α−γ>n−k−1,
so that all of them have sign (−1)n−kif α−γ>n−1.
4. Some examples of generalized linearization formulae
There are 18 different linearization formulae of the form (2) involving the three classical
families, which correspond to the expansion of the six possible products in series of each
family. Since we have just computed the complete set of connection coefficients for these
polynomials, the generalized linearization coefficientscan be conveniently evaluated bymeans
of (28). We shall illustrate how this formula works by means of some examples.
4.1. Expansions of products involving a Hermite polynomial in series of Hermite polynomials
We look for the coefficients of
pn(x)Hm(x ) =
n+m
X
k=0
gnmkHk(x) (52)
where Hm(x) denotes the monic Hermite polynomial of degree m. In this case, we trivially
have c(k−j,k)
m+j−k,r(q) =δm+j−k,r, so that (28) reduces to
gnmk =1
γk
j+
X
j=j−n
j m
k−jc(j,k)
n−j,m+j−k(p)~m+j−k,k
=2k−m
j+
X
j=j−n
j m
k−j(m +j−k)!
2jc(j,k)
n−j,m+j−k(p). (53)
If pn(x) =Hn(x ), then we also have c(j,k)
n−j,m+j−k(p) =δn−j,m+j−k. From (53) we readily
see that, in the (standard) linearization formula for Hermite polynomials,
Hn(x)Hm(x ) =
n+m
X
k=|n−m|
lnmkHk(x) (54)
the coefficient lnmk vanishes whenever n+m−kis odd, so that the sum over kin (54) can
be restricted to the values k=n+m−2rwith integer r. For such values of k, the only
non-vanishing term in the summation over jin the expression for lnmk given by (53) is that
corresponding to j=n−r, so that we have for k=n+m−2r,
lnmk =n
rm
rr!
2r.
Thus we obtain the well known Feldheim formula (cf [37], [1, p 195]),
Hn(x)Hm(x ) =
min(n,m)
X
r=0n
rm
rr!
2rHn+m−2r(x). (55)
14

The linearization coefficients in this formula are obviously positive, which has been found
useful in applications [38].
If pn(x) =L(α )
n(x), the connection coefficients c(j ,k)
n−j,m+j−k(p) in (53) correspond to the
expansion
L(α+j)
n−j(x) =
n−j
X
r=0
c(j,k)
n−j,r(p)Hr(x )
and the expression for these coefficients derived from (31) leads to
gnmk =(−1)n−m+k2k−m
j+
X
j=j−n
j m
k−j n−j
m+j−k(m +j−k)!(n +α+1)
m−n−k+2j
2j
×2F2m−n−k
2+j, m−n−k+1
2+j
m−k+α+1
2+j, m−k+α
2+j+1



1
4.(56)
We readily see that the sign of these coefficients is (−1)n−m+k. Likewise, when pn(x) =
P(α,β)
n(x), using equations (32) and (33) we can write gnmk as a double sum of terminating 2F0
or 2F1hypergeometric functions, which in the Gegenbauer case (α=β) reduce to a simple
sum of 2F0functions (cf (34)).
4.2. Linearization formulae for Laguerre polynomials
Let us consider now the linearization problem
L(α)
n(x)L(β )
m(x) =
n+m
X
k=0
gnmkL(γ )
k(x) (57)
where L(α)
n(x) denotes the monic Laguerre polynomial of degree nand parameter α; according
to (12), the linearization coefficients gnmk have the integral representation
gnmk =1
0(k +γ+1)k!Z∞
0L(α)
n(x)L(β )
m(x)L(γ )
k(x)x γe−xdx.
Equation (28) now reads
gnmk =1
0(k +γ+1)
j
+
X
j=j
−n
j m
k−jInmk(j)
Inmk(j) =
r+
X
r=0
c(j,k)
n−j,rc(k−j,k)
m+j−k,r0(r +k+γ+1)r!
(58)
where the connection coefficients correspond to the expansions
L(α+j)
n−j(x) =
n−j
X
r=0
c(j,k)
n−j,rL(γ +k)
r(x) L(β +k−j)
m+j−k(x) =
m+j−k
X
r=0
c(k−j,k)
m+j−k,rL(γ +k)
r(x).
From (38), we find that the explicit form of these coefficients is
c(j,k)
n−j,r =n−j
r(−1)n−j−r(α −γ+j−k)n−j−r
c(k−j,k)
m+j−k,r =m+j−k
r(−1)m+j−k−r(β −γ−j)m+j−k−r.
15

Substituting these expressions into (58), we obtain
gnmk =
j+
X
j=j−n
j m
k−jr+
X
r=0n−j
rm+j−k
rr!(k +γ+1)
r
×(γ −α+k−n+r+1)
n−j−r(γ −β+k−m+r+1)
m+j−k−r.(59)
We readily see that, if γ−αand γ−βare non-negative integers, then gnmk >0. This result
generalizes the non-negativity condition obtained by Koornwinder [34] for the particular case
when αand βare integers and γ=α+β. We also see from (59) that, if γ−α(resp. γ−β)
is not an integer, the non-negativity of the linearization coefficients still holds provided that
γ−α>n−1 (resp. γ−β>m−1). On the other hand, if α−γ>k−j
−and β−γ>j
+
,
then the sign of gnmk is (−1)n+m−k, so that all the linearization coefficients have this sign if
α−γ>mand β−γ>n.
The summation over rin (59) can be expressed as a 3F2hypergeometric function of unit
argument, which leads to
gnmk =
j+
X
j=j−n
j m
k−j(γ −α+k−n+1)
n−j(γ −β+k−m+1)
m+j−k
×3F
2j−n, k −j−m, k +γ+1
γ−α+k−n+1,γ −β+k−m+1


1
.(60)
In the particular case when γ=α+β, the expression for the linearization coefficients given
by the previous formula can be further simplified by taking advantage of the Pfaff–Saalsch¨
utz
summation theorem (A.9), which yields
gnmk =
j+
X
j=j−n
j m
k−j(k −m+n−j+α+1)
m+j−k(m −n+j+β+1)
n−j
=m
k(n −m+k+α+1)
m−k(m −n+β+1)
n
×3F
2−n, −k, m −n−k−α
m−k+1,m−n+β+1


−1
.(61)
We already know, from the discussion after (59), that gnmk >0ifα, β ∈Z, while if α(resp.
β) is not an integer the linearization coefficients are still non-negative provided that α>m−1
(resp. β>n−1). Equation (61) enables us to improve on the latter result, since inspection
of its right-hand side shows that, if α(resp. β) is not an integer, the non-negativity of the
linearization coefficients holds under a less restrictive condition, namely that α>m−n−1
(resp. β>n−m−1); in particular, the coefficients are non-negative in the m=ncase.
Now let us turn to (59). Interchanging the order of summation and shifting the index jto
l=n−r−j, this formula can be written as
gnmk =m!r+
X
r=0n
r(k +γ+1)
r(γ −β+k−m+r+1)
m+n−k−2r
(m +n−k−2r)!0(k −n+r+1)
×3F
2r−n, k −m−n+2r, γ −α+k−n+r+1
β−γ−n+r, k −n+r+1 


−1
.(62)
In the particular case when α=β=γ, the 3F2hypergeometric function in the right-hand
side of this equation reduces to a 1F0one, which can be evaluated in closed form by means
16

of (A.3). Thus we find that the solution of the standard linearization problem for Laguerre
polynomials,
L(α)
n(x)L(α )
m(x) =
n+m
X
k=|n−m|
lnmkL(α)
k(x) (63)
is given by
lnmk =m!r+
X
r=0n
r2m+n−k−2r(k +α+1)
r(k −m+r+1)
m+n−k−2r
(m +n−k−2r)!0(k −n+r+1)
=2
m+n−kn!m!
(m +n−k)!0(k −n+1)0(k −m+1)
×3F
2k−m−n
2,k−m−n+1
2,k+α+1
k−n+1,k −m+1 


1
.(64)
These coefficients are non-negative, as follows from the discussion after (59); this result is a
particular case of a rather general non-negativity theorem for integrals of products of Laguerre
polynomials of the same parameter, which is related to the combinatorial interpretation of
these integrals [12, lecture six]. It is worth noting that, while alternative expressions for the
Laguerre linearization coefficients in the cases γ=α+βand α=β=γcan be found in the
literature [16,37], the remarkably compact expressions (61) and (64) appear to be new. Let us
also note that the hypergeometric function in (64) can be evaluated in closed form by means
of the Pfaff–Saalsch¨
utz formula (A.9) in the case α=−1
2.
5. Conclusions
In this work, we have described a method to solve the general hypergeometric linearization
problem, i.e. the expansion of products of two arbitrary continuous hypergeometric-type
polynomials in terms of a sequence of orthogonal hypergeometric polynomials. Our approach
allows us to find integral representations for the associated linearization and connection
coefficients, in terms of the coefficients of the differential operators corresponding to the
involved polynomials, which are suitable for symbolic manipulation. To illustrate the method,
wehave found the explicit expressions of connection and linearization coefficients for thethree
classical families with real orthogonality (Hermite, Laguerre and Jacobi). These coefficients
are generally given in the form of terminating hypergeometric series, which at times can be
evaluated in closed form by means of classical summation theorems. In several cases, we have
been able to obtain sign properties such as positivity or non-negativity conditions from the
explicit representations found for the coefficients.
Itis worth noting that an affine transformation of the variablepreserves thehypergeometric
character of the polynomial families, so that our method is also applicable in these cases.
Furthermore, the present approach can be extended straightforwardly to hypergeometric
polynomials in a discrete variable [39], as well as to q-polynomials [40]. It is complementary
to the recursive approach [15], which supplies the linearization coefficients recurrently but
makes use of two or more characterization properties of the involved polynomials.
In our opinion, our method is a good starting point on the long road to solving
the general problem of linearization of products of arbitrary special functions other than
hypergeometric-type polynomials. Particular cases of this general problem corresponding to
Bessel functions, Whittaker functions, Jacobi functions, spheroidal wavefunctions and some
associated hypergeometric polynomials have been recently considered [28,41]. Some further
steps on the aforesaid road are the following, as yet unsolved, problems: the linearization
17

of basis-set functions, the expansion of products of special functions in terms of orthogonal
hypergeometric-type polynomials, the expansion of arbitrary special functions in terms of
products of two hypergeometric-type polynomials [42], the linearization of products of
two Nikiforov–Uvarov functions [5], the linearization and connection of two associated
hypergeometric-type polynomials, the determination of generating functions of products of
two hypergeometric-type functions [43], and the study of linear dependences among products
of basis-set functions [44]. Solutions to these problems would give us profound insight into
the algebraic properties of the special functions themselves, which would be very useful in
other branches of mathematics and applied science since, in particular, it would allow us to
gain insight into the matrix elements of the observables characterizing quantum mechanical
systems.
Acknowledgments
Thiswork was partially supported by research grants fromthe Direcci´
onGeneral de Ense˜
nanza
Superior (DGES) of Spain (project codes PB95-1205 for AMF and JSD, and PB96-0120 for
JSR), the European Union (INTAS-93-219-ext, for JSR, AMF and JSD), and the Junta de
Andaluc´
ıa (FQM0207 for JSR and JSD, FQM0229 for PLA and AMF).
Appendix A. Notations and formulae
A.1. Some special functions
We use the standard notations for the Gamma function, binomial coefficients and the
Pochhammer symbol, as well as their well known identities
(x)n=x(x +1)...(x +n−1)=0(x +n)
0(x) =(−1)n0(1−x)
0(1−x−n)
z
k=(−1)k(−z)k
k!0(2z) =22z−10(z +1
2)0(z)
0(1
2).
(A.1)
Assumingnto be a non-negative integer, we readily see from the definition of the Pochhammer
symbol that (x)n>0ifx>0, while the sign of (x)nis (−1)nfor x<1−n. On the other
hand, if 1 −n6x60, (x)n=0ifx∈Z, while otherwise its sign is (−1)[1−x].
The generalized hypergeometric function pFqis defined as
pFqa1,a
2,...,a
p
b
1,b
2,...,b
q



x=∞
X
k=0
(a1)k(a2)k...(a
p)
k
(b1)k(b2)k...(b
q)
k
xk
k!.(A.2)
In the simple case when p=1, q=0, Newton’s binomial theorem states that
1F0a
−



x=(1−x)−a.(A.3)
For the Gauss hypergeometric function (p=2, q=1) we have the special values (see,
e.g., [18]),
2F1−n, b
c


1=(c −b)n
(c)n
(A.4)
(Chu–Vandermonde sum),
2F1−n, c
2c


2=




0ifnis odd
(1
2)n/2
(c +1
2)n/2if nis even (A.5)
18

and
2F1a, b
a+b+1
2



1
2=0(1
2)0( a+b+1
2)
0(a+1
2)0( b+1
2).(A.6)
We also have the useful integration formula (see, e.g., [18, p 69]),
Zb
a
(x −a)µ−1(b −x)ν−1(cx +d)γdx
=(b −a)µ+ν−1(ac +d)γ0(µ)0(ν)
0(µ +ν) 2F1−γ,µ
µ+ν



c(a −b)
ac +d.(A.7)
Finally, two important results concerning the 3F2hypergeometric function are the classical
Watson’s summation theorem (see, e.g., [1, section 4.4] or [4, section 5.2.4]),
3F2a, b, c
a+b+1
2,2c


1=0(1
2)0(c +1
2)0( a+b+1
2)0( 1−a−b
2+c)
0(a+1
2)0( b+1
2)0( 1−a
2+c)0(1−b
2+c) (A.8)
and the Pfaff–Saalsch¨
utz formula (see, e.g., [1, p 66]),
3F2a, b, −n
d,a +b−n−d+1


1
=(d −a)n(d −b)n
(d)n(d −a−b)n
.(A.9)
The previous summation formulae hold whenever the hypergeometric series in the left-hand
sideare eitherterminating (herewe alwaysassume theparameter ntobe a non-negativeinteger)
or convergent; a detailed account of the validity conditions of each theorem can be found in
the indicated references.
A.2. Classical hypergeometric polynomials
We deal with the three classical families of monic hypergeometric polynomials orthogonal on
the real axis: Hermite, Laguerre and Jacobi, with their standard notation. In particular, we use
the following explicit formulae (see, e.g., [45]):
•Hermite polynomials:
Hn(x) =
[n/2]
X
k=0n
2k(−1
4)k(2k)!
k!xn−2k=xn2F0−n
2,1−n
2
−


−1
x2.(A.10)
•Laguerre polynomials:
L(α)
n(x) =(−1)n
n
X
k=0n
k(k +α+1)
n−k(−x)k
=(−1)n(α +1)
n1F
1−n
α+1



x
α>−1.(A.11)
•Jacobi polynomials:
P(α,β)
n(x) =
n
X
k=0n
k2n−k(k +α+1)
n−k
(n +k+α+β+1)
n−k
(x −1)k
=2n(α +1)
n
(n +α+β+1)
n2F
1−n, n +α+β+1
α+1 



1−x
2α, β > −1.
(A.12)
In the particular case when α=β, Jacobi polynomials are called Gegenbauer or
ultraspherical polynomials. In turn, some especially important particular cases of
19

Table A1. General data of the three classical families of monic orthogonal polynomials on the real
axis.
pn(x) Hn(x ) L(α)
n(x) P (α,β)
n(x)
(a, b) (−∞,∞)(0,∞)(−1,1)
σ(x) 1x1−x2
τ(x) −2xα+1−xβ−α−(α +β+2)x
ωk(x) e−x2xα+ke−x(1−x)α+k(1+x)β+k
γk√π0(k+α+1)2
2k+α+β+10(k+α+1)0(k+β+1)
0(2k+α+β+2)
An,k (−2)−n(−1)n(−1)n
(n+2k+α+β+1)n
pn,k(x ) Hn(x) L(α +k)
n(x) P (α+k,β+k)
n(x)
Gegenbauer polynomials are the Legendre polynomials (α=β=0), Chebyshev
polynomials of the first kind (α=β=−1
2), and Chebyshev polynomials of the second
kind (α=β=1
2).
All the necessary data concerning these families of polynomials (see, e.g., [1]) are gathered in
table A1.
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