Generalized Hermite polynomials

16 Pages·0.2 MB·English

by  Vadim V. Borzov

GENERALIZED HERMITE POLYNOMIALS 1
V.V. Borzov
Department of Mathematics, St.Petersburg University of Telecommunications,
191065, Moika 61, St.Petersburg, Russia
1 The new method for obtaining a variety of extensions of Hermite polynomials is given.
0
As a first example a family of orthogonal polynomial systems which includes the general-
0
ized Hermite polynomials is considered. Apparently, either these polynomials satisfy the
2
differential equation of the second order obtained in this work or there is no differential
n
a equation of a finite order for these polynomials.
J

5 KEYWORDS:orthogonalpolynomials,generalizedoscillatoralgebras,generalizedderiva-
2 tion operator.

] MSC (1991): 33C45, 33C80, 33D45, 33D80
A
Q
1. Introduction
.
h
t Inourformerwork([1])weconstructedanappropriateoscillator algebraA correspond-
a µ
m ing to the system of polynomials which are orthonormal with respect to a measure µ in
[ the space Hx = L2(R1;µ(dx)). By a standard manner the energy operator (hamiltonian)
1 Hµ = Xµ2 +Pµ2 was defined. The position operator Xµ was introduced by the recurrent
relations of the given polynomials system; the momentum operator P was determined as
v µ
6 an unitary equivalent to the position operator Y in the dual space H = L2(R1;ν(dy)).
µ y
1 In ([1]) it was proved that the usual differential equations for the classical polynomials
2
are equivalent to the equations of the form H ψ = λ ψ , where the eigenvalues of the
1 µ n n n
0 corresponding hamiltonian Hµ denote by λn. The central problem with a derivation of
1 the differential equations was findinga representation of the annihilation operator a (or
µ−
0 another ”reducing” operator) of the algebra A by a differential operator in the space H .
/ µ x
h Unfortunately, these formulas ([2]) are rather complicated in the general case. Therefore
t
a our interest is in describing such orthogonal polynomials systems for which appropriate
m
representations are simple. On this basis one can obtain some differential equations of
: a finite order (it is desirable that we have to deal only with differential equations of the
v
i second order).
X
The results of this work may be thought of as a first step forward in this direction.
r
a From our point of view we consider a family of orthogonal polynomial systems which
includes the generalized Hermite polynomials ([3]). These polynomials have been studied
extensively in the monograph ([4]). Therefore the polynomials of the considered family is
called Hermite-Chihara polynomials.
The paper is organized as follows. A generalized derivation operator is introduced in
Sec.2. By these operators a family of the Hermite-Chihara polynomials is determined
in Sec.3. More exactly the annihilation operator a of the algebra A corresponding to
−µ µ
the system of the polynomials may be represented by a generalized derivation operator
D . This operator is defined by a positive sequence ~v. In what follows we shall call this
~v
sequence ~v a ”governing sequence”. In Sec.4 we construct the generators X , P and H
µ µ µ
of the algebra A corresponding to a system of the Hermite-Chihara polynomials. As
µ
an example of such polynomials we consider the ”classical” Hermite-Chihara polynomi-
als ([4]) in Sec.5. Moreover, in this section a new derivation of the well-known ([3],[4])
1This research was supported by RFFIgrant No00-01-00500
1
2 V.V.BORZOV
differential equation for these polynomials is presented. Further, in Sec.6 we introduce a
special family of orthogonal systems of Hermite-Chihara polynomials which includes the
classical Hermite-Chihara polynomials. Furthermore, in this section we construct a ”gov-
erning sequence” ~v of an appropriate generalized derivation operator D . Then in Sec.7
~v
we obtain a differential equation of the second order for above-mentioned polynomials by
analogy with the derivation of the differential equation given in Sec.5. Finally, in the
conclusion we consider the following conjecture. If the polynomials of a system of orthog-
onal Hermite-Chihara polynomials satisfy a differential equation of the second order, then
these polynomials belong to the special family of orthogonal systems of Hermite-Chihara
polynomials introduced in Sec.6. Moreover, the other Hermite-Chihara polynomials do
not satisfy any differential equation of a finite order.
2. Generalized derivation operator
In this section we introduce a new class of differential operators (they are the infinite
order in general case) which play a large role in the construction of the Hermite-Chihara
polynomials. Let ~v = {vn}∞n=0 be a monotone nondecreasing sequence:
1= v v v v .... (2.1)
0 1 2 n
≤ ≤ ≤ ··· ≤ ≤
This sequence ~v define a linear operator D by the relations:
~v
D x0 = 0, D xn = v xn 1, n= 1,2..., (2.2)
~v ~v n 1 −
−
on the set of the formal power series of real argument x.
We will seek for the operator D of the type
~v
∞ dm
D = a xn . (2.3)
~v nm dxm
n,m=0
X
Substituting (2.3) in (2.2), we get the following formula:
∞ dk
D = ε xk 1 , (2.4)
~v k − dxk
k=0
X
The coefficients {εk}∞k=0 are defined by the recurrent relations:
v ε ε
ε1 = v0 = 1, εk = k−1 εk 1 k−2 1 , k = 1,2.... (2.5)
k! − − − 2! −···− (k 1)!
−
Definition 2.1. A differential operator D determined by formulas (2.4),(2.5) is called a
~v
generalized derivation operator induced of the sequence ~v.
Lemma 2.2. For the order of a generalized derivation operator D defined by formulas
~v
(2.4),(2.5) to be finite it is necessary and sufficient that the following equalities:
ε = ε = = 0. (2.6)
k+1 k+2
···
was valid.
To take three examples of generalized derivation operators of a finite order.
1. Let k = 1. There exist a unique solution of the system (2.6):
~v = {n+1}∞n=0. (2.7)
The generalized derivation operator D corresponding to ~v take the following form:
~v
d
D = . (2.8)
~v
dx
GENERALIZED HERMITE POLYNOMIALS 3
2. Let k = 2 and let v to be a number such that v 1. There exist a one-parameter
1 1
≥
family of the solution ~v = {vn}∞n=0 of the system (2.6):
v = 1, v = C2 v n2+1, n= 1,2.... (2.9)
0 n n+1 1−
The generalized derivation operator D corresponding to ~v take the following form:
~v
d v d2
1
D = +x( 1) . (2.10)
~v dx 2 − dx2
If v = 4, then from (2.9),(2.10) we have
1
d d2
~v = (n+1)2 ∞n=0, D~v = dx +xdx2 . (2.11)
3. Let k = 3 and let v (cid:8),v to be(cid:9)some number such that 1 v v . There exist a
1 2 1 2
≤ ≤
two-parameter family of the solution ~v = {vn}∞n=0 of the system (2.6):
(n+1)n(n 2)
v = 1 v , v = C3 v − v +
0 ≤ 1 n n+1 2− 2 1
(n+1)(n 1)(n 2)
+ − − , n= 1,2.... (2.12)
2
The generalized derivation operator D corresponding to ~v take the following form:
~v
d v d2 v 3v +3 d3
D = +x( 1 1) +x2 2− 1 . (2.13)
~v dx 2 − dx2 3! dx3
If v = 8, v = 27, then from (2.12) and (2.13) we have
1 2
d d2 d3
~v = (n+1)3 ∞n=0, D~v = dx +xdx2 +x2dx3 . (2.14)
(cid:8) (cid:9)
3. Hermite-Chihara polynomials
Let µ be a symmetric probability measure, i.e. the all odd moments of the measure
µ are vanish and ∞ µ(dx) = 1. In this section we consider a system of polynomials ,
which are orthonorm−∞al with respect to the measure µ , such that there is a representation
R
of the annihilation operator of the oscillator algebra A corresponding to this system by
µ
a generalized derivation operator.
Recall ([1]) that the recurrent relations of a canonical orthonormal polynomials system
{ψn(x)}∞n=0 take the following form:
xψ (x) = b ψ (x)+b ψ (x), n 1, (3.1)
n n n+1 n 1 n 1
− − ≥
x
ψ (x) = 1, ψ (x) = . (3.2)
0 1
b
0
In ([1])it was described how to get the positive sequence {bn}∞n=0 from the given sequence
{µ2n}∞n=0 of even moments of a symmetric positive measure µ.
The question we are interested now is when for a canonical orthonormal polynomials
system {ψn(x)}∞n=0 there are two sequences such that:
1. a positive sequence ~v = {vn}∞n=0 which satisfies (2.1);
2. a real sequence ~γ = {γn}∞n=0 for which are hold the following relations:
D ψ = 0, D ψ = γ ψ , n = 1,2..., (3.3)
~v 0 ~v n n n 1
−
where the generalized derivation operator D is determined by formulas (2.4),(2.5).
~v
We denote by [n] the following symbol:
b2
[0] = 0, [n]= n−1, n = 1,2.... (3.4)
b2
0
4 V.V.BORZOV
Let J be a symmetric Jacobi matrix
J = {bij}∞i,j=0
whichhasthepositiveelements b = b , i = 0,1,... onlydistinctfromzero. Then
i,i+1 i+1,i
the polynomials of the first kind can be represented in the form ([5]):
ǫ(n)
2 ( 1)m
ψn(x) = − b20m−nα2m 1,n 1xn−2m, (3.5)
[n]! − −
m=0
X
where the greatest integer function ispdenoted byǫ(α) . The coefficients α for any
2m 1,n 1
n 1, ǫ(n) m 1 are defined by the following equalities: − −
≥ 2 ≥ ≥
n−1 k1−2 km−1−2
α = 0, α = [k ] [k ] [k ]. (3.6)
1,n 1 2m 1,n 1 1 2 m
− − − − ···
k1=X2m−1 k2=X2m−3 kXm=1
Substituting (3.6),(3.5) and (3.4) into (3.3), it is easy to prove the following theorem ([2]).
Theorem 3.1. Lettheorthonormal polynomial system{ψn(x)}∞n=0 isdefinedby(3.6),(3.5)
and (3.4). For existence two sequences ~v = {vn}∞n=0 and ~γ = {γn}∞n=0 such that the con-
ditions (3.3) are hold it is necessary and sufficient that
1. the sequence ~v = {vn}∞n=0 satisfies (2.1) and the following conditions:
v v +v v = v v +v v , (3.7)
n 2 2p 1 2p 3 n 2p n 2p 3 2p 1 n 2p
− − − − − − −
for any n 2, 2p n;
≥ ≤
2. the coefficients α take the following form
2m 1,n 1
− −
[2m 1]!!(v )!
n 1
α2m 1,n 1 = − − , (vk)! = v0v1 vk, (3.8)
− − (v2m 1)!(vn 2m 1)! ···
− − −
as n 1, 2m n and regarding (v )!= (v )! = 1.
1 0
In≥this case th≤e sequence ~γ = {γn}−∞n=0 is defined by the following formulas:
v v
1 n 1
γn = b2(v −v ), n ≥ 1. (3.9)
r 0 n− n−2
Here we will not given the proof of this theorem (see [2]) to save room. However
we present some formulas arising from the proof. These expressions relate the sequence
~γ = {γn}∞n=0 and the coefficients α2m−1,n−1:
1 [2]
γ = , b γ = ε +ε , (3.10)
1 0 2 1 2
b 2
0 p
α
2p 1,2p
[2p+1]γ2p+1 = − γ1, (3.11)
[2p 1]!!
−
p α
2p 1,2p+1
[2p+2]γ2p+2 = − [2]γ2, (3.12)
α
2p 1,2p
p − p
where ε and ε are defined by (2.5).
1 2
Now we shall give the following definition.
Definition 3.2. The orthonormal polynomials system {ψn(x)}∞n=0 completed in Hx =
L2(R;µ(dx)) is called a system of Hermite-Chihara polynomials if these polynomials are
defined by (3.6),(3.5) and (3.4).
GENERALIZED HERMITE POLYNOMIALS 5
Remark 3.3. It is clear that the Hermite polynomials fall in this class. Here
d n
~v = {n+1}∞n=0, D~v = dx, b2n−1 = 2, [n] = n, n ≥ 1.
According to (3.8), we have
n!
α = . (3.13)
2m−1,n−1 2mm!(n 2m)!
−
Substituting (3.13) into (3.5), we obtain the usual form of the Hermite polynomials (see
[1], [3] ). In addition, γ = √2n, n 1 and then (3.3) is reduced to the usual rule of
n
≥
derivation for the Hermite polynomials:
d
H (x) = 2nH (x) . (3.14)
n n 1
dx −
Remark 3.4. According to the theorem 3.1, by any sequence~v = {vn}∞n=0 complying with
(2.1) and (3.7) we can write the the coefficients α which take the following form:
2m 1,n 1
− −
v (v v )
n 1 n n 2
[1] = 1, [n]= − − − , n 2, (3.15)
v ≥
1
v (v v )
b2n−1 = b20 n−1 nv1− n−2 , n ≥ 2. (3.16)
The polynomials ψ (x) satisfy the recurrent relations (3.1) and (3.2). By solving the
n
Hamburger moment problem of the Jacobi matrix J, we obtain the symmetric probability
measure µ such that the polynomials of the system {ψn(x)}∞n=0 are orthonormal with
respect to µ. If the moment problem for the Jacobi matrix J is a determined one, then
the measure µ is defined uniquely. Otherwise (when the moment problem for the Jacobi
matrix J is a undetermined one) there is a infinite family of such measures (see [6]).
4. Oscillator algebra for the Hermite-Chihara polynomials
In this section we constructthe generalized Heisenberg algebra A correspondingto the
µ
system of the Hermite-Chihara polynomials (see [1]).
Let ~v = {vn}∞n=0 be the positive sequence such that the conditions (2.1) and (3.7) are
hold. Then the sequence {bn}∞n=0 can be found by (3.16). Furthermore, we obtain the
system of the Hermite-Chihara polynomials {ψn(x)}∞n=0 by the formulas (3.5) and (3.8).
These polynomials satisfy the recurrent relations (3.1) and (3.2) with above-mentioned
coefficients {bn}∞n=0. Under the condition
∞ b 1 = v1 ∞ 1 = . (4.1)
nX=0 −n b0 nX=1 vn(vn+1−vn−1) ∞
the moment problem for the correspondipng Jacobi matrix is a determined one (see [6]).
There is the only symmetric probability measure µ such that the polynomials {ψn(x)}∞n=0
areorthonormalinthespaceH = L2(R;µ(dx)). Inaddition,theevenmomentsµ ofthe
x 2n
measure µ can be found from the following algebraic equations system (b = 0, n 0)
1
− ≥
ǫ(n) ǫ(n)
2 2 ( 1)m+s
− α α µ = b2 +b2. (4.2)
(bn 1)! 2m−1,n−1 2s−1,n−1 2+2n−2m−2s n−1 n
mX=0mX=0 −
Itis easy to check that thecondition (4.1) is correct for the ”classical” Hermite-Chihara
polynomials to be considered in the next section.
6 V.V.BORZOV
From theorem 3.1 it follows that there are the generalized derivation operator D de-
~v
termined for given sequence ~v by formulas (2.4),(2.5) and the sequence ~γ = {γn}∞n=0:
v
n 1
γn = − , n 1, (4.3)
b ≥
n 1
−
such that the relations (3.3) are valid .
Using the methods of ([1]), we construct the generalized Heisenberg algebra A corre-
µ
sponding to the ortonormal system {ψn(x)}∞n=0. By the usual formulas we define ladder
operators a (the annihilation operator), a+ (the creation operator) and the number op-
−µ µ
erator N. It is readily seen that
a = D f(N), (4.4)
−µ ~v
The operator-function f(N) acts on basis vectors {ψn(x)}∞n=0 by formulas:
v v
f(N)ψ0 = 0, f(N)ψ1 = √2b20ψ1, f(N)ψn = √2b20 n −v n−2ψn, (4.5)
1
where n 2. The position operator X is defined by the recurrence relations (3.1) and
µ
≥
(3.2). Using X and a , we determine by the well-known formulas (see [1]) the operators
µ −µ
a+, P (the momentum operator) and H (hamiltonian):
µ µ µ
P = ı(√2a X ), a+ = √2X a , (4.6)
µ −µ µ µ µ −µ
− −
H = X2+P2 = √2(a X +X a+) = a a++a+a . (4.7)
µ µ µ −µ µ µ µ −µ µ µ −µ
We have the following commutation relation:
[a ,a+] = 2(B(N +I) B(N)). (4.8)
−µ µ
−
The operator-function f(N) acts on basis vectors by formulas:
v (v v )
B(N)ψ0 = 2b20, B(N)ψn = b2n−1ψn = b20 n−1 nv1− n−2 ψn, n ≥ 1. (4.9)
Moreover, the ”energy levels” are
2b2
λ = 2b2, λ = 2(b2 +b2) = 0(v v v v ), (4.10)
0 0 n n−1 n v1 n n+1− n−1 n−2
where n 1.
≥
In what follows our prime interest is with the following question. Is it possible to
get a differential equation of the second order for Hermite-Chihara polynomials from the
equation H ψ = λ ψ .
µ n n n
5. Classical Hermite-Chihara polynomials
Now we consider a particular case of the Hermite-Chihara polynomials, namely, the
well-known (see [3]) generalized Hermite polynomials which have been studied extensively
in ([4]) (see also ([8])).
We denote by H the Hilbert space
γ
1
H = L2(R; xγ(Γ( (γ +1))) 1exp( x2)dx), γ 1. (5.1)
γ −
| | 2 − ≥ −
Ucosminpglemteedthiondsthoef(s[p1]a)c,ewHec.onTsthreucptotlhyenocamnioanlsicψalo(xrt)hsoantoisrfmyatlhpeolryencoumrrieanltssryeslatetmion{sψ(n3(.x1))}∞n=0
γ n
GENERALIZED HERMITE POLYNOMIALS 7
and (3.2). The coefficients {bn}∞n=0 are defined by formulas (3.16), where b0 = γ+21 and
the sequence ~v = {vn}∞n=0 is given by the following equalities: q
γ+n+1 n = 2m,
v = γ+1 (5.2)
n n+1 n = 2m+1.
( γ+1
It is clear that v0 = 1, v1 = γ+21 = b−02. The coefficients {bn}∞n=0 are defined by the
formulas:
1 √n n = 2m,
b = (5.3)
n−1 2( √n+γ n = 2m+1.
The formulas (3.8),(3.5),(3.4) give a explicit form of the polynomials ψ (x). Recall that
n
the polynomials
Kγ(x) = s ψ (x), n 0,
n n n
≥
as s = 1 and s = (b )!, are named the generalized Hermite polynomials in ([4])(
0 n n 1
−
see also ([8])). In what follows we shall call the polynomials ψ (x) as the ”classical
n
Hermite-Chihara polynomials”. It is easy to prove that the family of these polynomials is
a particular case of the more general class of Hermite-Chihara polynomials and that the
generalized derivation operator D corresponding to the given sequence ~v is determined
~v
by formulas (2.4) and (2.5), where
( 2)m 1 γ
−
ε =1, ε = − , m 2. (5.4)
1 m
m! γ+1 ≥
In addition, the sequence ~γ = {γn}∞n=0 appearing in (3.3) is defined by equalities:
√2 √n n= 2m,
γ = (5.5)
n γ +1 √n+γ n = 2m+1.
(
Comparing (4.4), (4.5) and (3.3), we obtain
γ+1
a = D . (5.6)
−µ √2 ~v
The following formulas are known ([4])( see, also([8])):
d d n (n 1)θ
ψ0 = 0, ψn = ψn 1+ − n X−1ψn 2 = (5.7)
dx dx bn 1 − 2bn 1bn 2 −
− − −
θ
n
= 2b ψ ψ , n 1, (5.8)
n 2 n 1 n
− − − x ≥
where
1 ( 1)n
θ = θ (γ) = γ − − . (5.9)
n n
2
Taking into account how the annihilation operator a and the number operator N act on
−µ
the basis vectors {ψn(x)}∞n=0, it is easy to get from the relations (5.7)-(5.9) the following
formula:
d
X N = (a )2. (5.10)
µdx − −µ
Note also that the action of the position operator Xµ on the basis vectors {ψn(x)}∞n=0 is
defined by (3.1) and (5.3). Now we consider the operator
Θ =2B(N) N, (5.11)
N
−
8 V.V.BORZOV
wheretheoperator-function B(N) is definedby (4.10). Using(5.9) and(5.11), wesee that
Θ ψ = θ ψ , n 1. (5.12)
N n n n
≥
Taking intoaccount theequation H ψ = λ ψ , whereahamiltonian H definedby(4.7),
µ n n n µ
and the equality a+a = 2B(N), we have the following relation:
µ −µ
a a+ = 2B(N +I). (5.13)
−µ µ
Moreover, from (5.8) it follows that
1 d 1
a = + X 1Θ . (5.14)
−µ √2dx √2 µ− N
Now from (5.13), (5.14) and (4.7) we have
d 1 d 1
( +X 1Θ )(X X 1Θ )= 2B(N +I). (5.15)
dx µ− N µ − 2dx − 2 µ− N
Multiplying both sides of (5.15) by 2X from the left, we obtain:
µ
−
d d2
2X (I +X )+X +X ( X 2Θ +X 1Θ ) (5.16)
− µ µdx µdx2 µ − µ− N µ− N
d
2Θ X +Θ +Θ X 1Θ = 2X 2B(N +I). (5.17)
− N µ Ndx N µ− N − µ
It is not hard to prove that :
Θ X 1Θ ψ = 0, Θ X = X Θ , (5.18)
N µ− N n N µ µ N+I
2X (2B(N +I) Θ ) 2X = 2X N, (5.19)
µ N+I µ µ
− −
d d
(Θ + Θ )ψ = γψ . (5.20)
Ndx dx N n n′
Applyingbothsidesof(5.17)toψ andusing(5.18)-(5.20), wegetthefollowingdifferential
n
equation
θ
xψ +(γ 2x2)ψ +(2nx n)ψ = 0, n 0, (5.21)
n′′ − n′ − x n ≥
which is coincident with the well-known differential equation for the classical Hermite-
Chihara polynomials [4] (see also [3]).
Remark 5.1. The generators a+,a of the generalized Heisenberg algebra A correspond-
µ −µ µ
ing to the classical Hermite-Chihara polynomials system subject to the following commu-
tative relation (see [1]):
[a ,a+] = (γ +1)I 2Θ . (5.22)
−µ µ N
−
The ”energy levels” of the associated oscillator are equal to:
λ = γ +1, λ = 2n+γ+1, n 1. (5.23)
0 n
≥
Finally, it follows from (4.7) and (5.14) that the momentum operator take the following
form:
d
P = ı( +X 1Θ X ). (5.24)
µ dx µ− N − µ
GENERALIZED HERMITE POLYNOMIALS 9
6. Construction of the ”governing sequence” ~v of a generalized
derivation operator D for the classical Hermite-Chihara polynomials
~v
Let {ψn(x)}∞n=0 be a orthonormal Hermite-Chihara polynomials system. According
to theorem 3.1, there is a sequence ~v such that (2.1) and (3.7) are hold. The formula
(3.9)allows us to define the sequence ~γ by ~v. Furthermore, the generalized derivation
operator D~v corresponding to~v is a reducing operator for the system {ψn(x)}∞n=0, i.e. the
equalities (3.3) are valid. In this section we obtain the exact condition on ~v which select
some family of Hermite-Chihara polynomials. This family is a natural extension of the
set of classical Hermite-Chihara polynomials. The associated set of~v is a three-parameter
family dependingon theparameters b ,v andv . Butit turnout thatthe parameter v is
0 1 2 1
unessential, so that the above-mentioned family is really a two-parameter one. According
to (2.4) and (2.5)), the generalized derivation operator D corresponding to a sequence ~v
~v
complying with (2.1) and (3.7) take the following form:
d d
D = X 1(B +B )= +X 1B , B = X , (6.1)
~v − 1 1 − 1 1
dx dx
where
∞ dk
B = ε xk . (6.2)
1 k dxk
k=2
X
The coefficients ε in (6.2) are defined from given sequence ~v by the recurrent relations
k
(2.5).
Remark 6.1. For the classical Hermite-Chihara polynomials it follows from(6.2) and (5.4)
that
d
cl
X +B = δ(N), (6.3)
1
dx
where δ(N) is the projection on the subspace of the polynomials of odd degree:
δ(N)xn =θ (1)xn, n 0,
n
≥
(see (5.9)) and hence
δ(N)ψcl = θ (1)ψcl.
n n n
We denote by {ψn(x)}∞n=0 the orthonormal Hermite-Chihara polynomials system which
is constructed according to remark 3.4 for given sequence ~v. Obviously,
B ψ = B ψ = 0.
1 0 1 1
Now we shall restrict our consideration to a particular class of the Hermite-Chihara
polynomials for which there are two real sequence ~δ = δn ∞n=2 and β~ = βn ∞n=0 such
that:
(cid:8) (cid:9) (cid:8) (cid:9)
B ψ = δ Xψ , B ψ = δ Xψ +β ψ , n 3. (6.4)
1 2 2 1 1 n n n 1 n n 2
− − ≥
Replacing δ by δ and β by β , we see from (6.1) and (3.3) that the condition (6.4) is
n n n n
valid for B too. Then it is clear that the assumption (6.4) takes the place of the rule of
1
derivation (5.8). Substituting (6.2) into (6.4), and taking into account (3.5), we obtain
10 V.V.BORZOV
the following relation:
n ǫ(n−2k)( 1)m (n m)!
−[n]! b20m−nα2m−1,n−1xn−2m(n 2−m k)! =
k=2 m=0 − −
X X
pǫ(n−2)
2 ( 1)m
= βn − b20m−n+2α2m 1,n 3xn−2m−2+
[n 2]! − −
mX=0 −
ǫ(n−p1)
2 ( 1)m
+δn [−n 1]!b20m−n+1α2m−1,n−2xn−2m, n≥ 2. (6.5)
mX=0 −
Equating the coefficients at xn ipn the both sides of (6.5), we get
δ b = A (2), n 2, (6.6)
n n 1 n
− ≥
where we used the notation
s
ε
k
A (m)= s! , s m, (6.7)
s
(s k)! ≥
k=m −
X
and the coefficients ε are defined by formulas (2.5). For the classical Hermite-Chihara
k
polynomials from (5.4), (5.5) and binomial formula it follows that (as n 2)
≥
cl √2γ √n n = 2m,
δ = (6.8)
n γ+1( √nn−+1γ n = 2m+1.
In order that to find the quantities β in (6.4) we equal the coefficients at xt in the both
n
sides of (6.5) (as 0 t < n). Obviously, a coefficients at xt only distinct from zero when
≤
it is valid the following condition:
n
n t = 2p, 1 p ǫ( ). (6.9)
− ≤ ≤ 2
We consider separately three cases t = 0,1,2.
1. Let t = 0, n =2p. We have β = 0, p 1.
2p
≥
2. Let t = 1, n =2p+1. We have
[2p]α
2p 3,2p 2
δ2p+1 = − − β2p+1, p 1. (6.10)
b α ≥
p 0 2p 1,2p 1
− −
3. Let t = 0, n 2 = 2p. Regarding p 1, we have
− ≥
α A (2) = β [2p+2][2p+1]α +
2p−1,2p+1 2 − 2p+2 2p−3,2p−1
p +α2p 1,2pA2p+2(2). (6.11)
−
Taking into account that β = 0, we have from here
2p
α A (2) α A (2) = 0, p 1. (6.12)
2p 1,2p 2p+2 2p 1,2p+1 2
− − − ≥
Using (3.8), we simplify this relation:
v A (2) v A (2) = 0, p 1. (6.13)
1 2p+2 2p+1 2
− ≥
From the equalities (2.5) and the designation (6.7) it follows that:
A (2) = v k, k 2. (6.14)
k k 1
− − ≥
Substituting (6.14) into (6.13), we get
v = (p+1)v , p 1. (6.15)
2p+1 1
≥