Resonant equations with classical orthogonal polynomials. II
UDC 517.9 We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite and the Laguerre orthogonal polynomials, and propose an algorithm of finding their particular and general solutions in the closed form. The algorithm is especially su...
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Institute of Mathematics, NAS of Ukraine
2019
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507209076572160 |
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| author | Gavrilyuk, I. P. Makarov, V. L. Гаврилюк, І. П. Макаров, В. Л. |
| author_facet | Gavrilyuk, I. P. Makarov, V. L. Гаврилюк, І. П. Макаров, В. Л. |
| author_sort | Gavrilyuk, I. P. |
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| description | UDC 517.9
We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite
and the Laguerre orthogonal polynomials, and propose an algorithm of finding their particular and general solutions in the
closed form. The algorithm is especially suitable for the computer-algebra tools, such as Maple. The resonant equations
form an essential part of various applications, e.g., of the efficient functional-discrete method for the solution of operator
equations and eigenvalue problems. These equations also appear in the context of supersymmetric Casimir operators for the di-spin algebra, as well as of the square operator equations $A^2u = f$ , e.g., of the biharmonic equation. |
| first_indexed | 2026-03-24T02:05:40Z |
| format | Article |
| fulltext |
UDC 517.9
I. Gavrilyuk (Univ. Cooperative Education Gera-Eisenach, Germany),
V. Makarov (Inst. Math. Nat. Acad. Sci. Ukraine, Kyiv)
RESONANT EQUATIONS WITH CLASSICAL
ORTHOGONAL POLYNOMIALS. II
РЕЗОНАНСНI РIВНЯННЯ З КЛАСИЧНИМИ
ОРТОГОНАЛЬНИМИ ПОЛIНОМАМИ. II
We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite
and the Laguerre orthogonal polynomials, and propose an algorithm of finding their particular and general solutions in the
closed form. The algorithm is especially suitable for the computer-algebra tools, such as Maple. The resonant equations
form an essential part of various applications, e.g., of the efficient functional-discrete method for the solution of operator
equations and eigenvalue problems. These equations also appear in the context of supersymmetric Casimir operators for
the di-spin algebra, as well as of the square operator equations A2u = f , e.g., of the biharmonic equation.
Вивчаються резонанснi рiвняння, що пов’язанi з класичними ортогональними многочленами, заданими на нескiн-
ченних iнтервалах, тобто з ортогональними многочленами Ермiта i Лагерра. Запропоновано алгоритм знаходження
їхнiх частинних розв’язкiв i загального розв’язку в замкненому виглядi. Цей алгоритм є особливо зручним в iм-
плементацiї засобами комп’ютерної алгебри, наприклад, Maple. Резонанснi рiвняння є вагомою складовою рiзних
застосувань, наприклад ефективного функцiонально-дискретного методу розв’язування операторних рiвнянь i за-
дач на власнi значення. Такi рiвняння виникають також у контекстi суперсиметричних операторiв Казимiра для
дi-спiнової алгебри, а також при розв’язуваннi операторних рiвнянь з квадратом деякого оператора, наприклад
бiгармонiчного рiвняння.
1. Introduction. This paper represents the second part of the eponymous paper from the previous
issue of this journal. Here we study the resonant equations with the differential operators defining
the classical orthogonal polynomials on infinity intervals, namely the Hermite and the Laguerre
orthogonal polynomials. We use the Algorithm 3.1 from part I (see [4]) to obtain the particular
solutions of the corresponding resonant equations of the first and of the second kind. We obtain
explicit formulas for the general solutions of the corresponding inhomogeneous resonant differential
equations.
2. Resonant equation of the Hermite type. 2.1. The Hermite resonant equation of the first
kind. In this section we consider the following resonant gather of the Hermite type:
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
x2
\bigr) d
dx
\biggl[
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- x2
\bigr) du(x)
dx
\biggr]
+ 2nu(x) = Hn(x), (2.1)
where Hn(x) is the Hermite polynomial, satisfying the homogeneous differential equation. The
Hermite polynomial Hn(x) can be represented through the hypergeometric function
H\nu (x) =
2\nu
\surd
\pi
\biggl(
1 - \nu
2
\biggr)
[n2 ]+1
\Gamma
\biggl( \Bigl[ n
2
\Bigr]
+ 1 +
1 - \nu
2
\biggr) 1F1
\biggl[
- \nu
2
;
1
2
;x2
\biggr]
-
2\nu +1\surd \pi
\biggl(
- \nu
2
\biggr)
[n2 ]+1
\Gamma
\Bigl( \Bigl[ n
2
\Bigr]
+ 1 - \nu
2
\Bigr) x 1F1
\biggl[
1 - \nu
2
;
3
2
;x2
\biggr]
(2.2)
for \nu = n (see [6, p. 147]). The general solution of the homogeneous equation (2.1) is given by
c\bigcirc I. GAVRILYUK, V. MAKAROV, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4 455
456 I. GAVRILYUK, V. MAKAROV
u(x) = c1Hn(x) + c2hn(x),
where
hn(x) = -
\infty \int
\infty
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- t2
\bigr)
Hn(t)
t - z
dt, n = 0, 1, 2, . . . , x \in \BbbC \setminus ( - \infty ,\infty ),
is the Hermite functions of the second kind, which satisfies the recurrence equation for the Hermite
polynomials. This function can be expressed also through the confluent hypergeometric function in
the following way [5]:
h2n(x) = ( - 1)n2n+1(2n)!!x1F1
\biggl(
- 2n - 1
2
;
3
2
;x2
\biggr)
=
=
\bigl[
p2n(x) \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
x2
\bigr)
+
\surd
\pi H2n(x)erfi(x)
\bigr]
, n = 1, 2, . . . ,
h2n+1(x) = ( - 1)n+1 (2n)!!2n+1
1 F1
\biggl(
- 2n+ 1
2
;
1
2
;x2
\biggr)
=
=
\bigl[
p2n+1(x) \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
x2
\bigr)
+
\surd
\pi H2n+1(x)erfi(x)
\bigr]
, n = 0, 1, . . . .
These formulas were obtained by Maple solving the Hermite differential equation and they satisfy
the difference equation
pn+1(x) = 2xpn(x) - 2npn - 1(x), n = 1, 2, . . . ,
p0(x) = 0, p1(x) = - 2.
(2.3)
The formulas for the odd and the even indexes can be unified in the following formula:
hn(x) = ( - 1)[
n+1
2 ]2[
n
2 ]+1
\Bigl(
2
\Bigl[ n
2
\Bigr] \Bigr)
!! x2\{
n+1
2 \} \times
\times 1F1
\biggl(
- n
2
+
\biggl\{
n+ 1
2
\biggr\}
;
1
2
+ 2
\biggl\{
n+ 1
2
\biggr\}
;x2
\biggr)
, n = 0, 1, . . . , (2.4)
where [x] and \{ x\} denote the integer and the fractional parts of the real number x.
The last expression can be transformed to
hn(x) = Hn(x)
\surd
\pi erfi(x) + pn(x) \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
x2
\bigr)
,
where the polynomials pn(x) satisfy the recurrence equation (2.3).
We use Theorem 3.1 of [4] to find a particular solution of the inhomogeneous equation. We begin
with the case n = 0, i.e., we differentiate representation (2.2) by \nu , i.e.,
\~u0(x) = - 1
2
d
d\nu
1F1
\biggl(
- \nu
2
;
1
2
;x2
\biggr) \bigm| \bigm| \bigm| \bigm|
\nu =0
,
set thereafter \nu = 0 and omit some summands, which satisfy the homogeneous equation
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. II 457
u0(x) =
1
4
\infty \sum
p=1
x2p
p
\biggl(
1
2
\biggr)
p
=
\surd
\pi
2
x\int
0
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
t2
\bigr)
[1 - erfc(t)] dt. (2.5)
Analogously in order to obtain u1(x) we set n = 1 in (2.2), differentiate by \nu , substitute \nu = 1
and omit some summands satisfying the homogeneous differential equation. Then we obtain with the
assistance of Maple
u1(x) = - x
d
d\nu
1F1
\biggl(
1 - \nu
2
;
3
2
;x2
\biggr) \bigm| \bigm| \bigm| \bigm|
\nu =1
=
1
2
x
\infty \sum
p=1
x2p
p
\biggl(
3
2
\biggr)
p
=
=
x
2
x\int
0
1
t2
\bigl\{ \surd
\pi \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
t2
\bigr)
[1 - erfc(t)] - 2t
\bigr\}
dt, (2.6)
where erfc(x) is the imaginary error function [2].
One can observe that this way to obtain particular solutions is very cumbersome. Below one can
see that Algorithm 3.1 from part I [4] provides a more comfortable way.
Actually, let us differentiate the recurrence relation for the Hermite polynomials
Hn+1(x) - 2xHn(x) + 2nHn - 1(x) = 0
by n, then using Theorem 3.1 of [4] we obtain the following recursion:
un+1(x) = 2xun(x) - 2nun - 1(x) +Hn - 1(x), n = 1, 2, . . . . (2.7)
Using (2.5), (2.6) we have the following expressions as particular solutions:
\chi 0(x) = E(x) =
\surd
\pi
2
x\int
0
erf(t) \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
t2
\bigr)
dt,
\chi 1(x) =
1\sum
p=0
( - 1)pCp
1H1 - p(x)
dp
dxp
E(x) + x.
Further we use the ansatzes
u0(x) = \chi 0(x) + c0, u1(x) = \chi 1(x) + c1x (2.8)
with undefined coefficients c0, c1 for the initial values of Algorithm 3.1 from part I (see [4]). Sub-
stituting these into the recurrence equation (2.7) with n = 1 and choosing these coefficients so that
u2(x) satisfies the resonant equation, we obtain that c0 can be arbitrary and c1 should satisfy the
equation
4 + 4c1 = 0,
i.e., c1 = - 1. Note that if we choose c0 = 0, then we arrive at the representation
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
458 I. GAVRILYUK, V. MAKAROV
un(x) =
n\sum
p=0
( - 1)pCp
nHn - p(x)
dp
dxp
E(x),
E(x) =
\surd
\pi
2
x\int
0
erf(t) \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
t2
\bigr)
dt
(2.9)
which for n = 0, 1, 2 was obtained in [3].
We have constructed uk(x), k = 0, 1, 2, so, that these functions are particular solutions of
the Hermite resonant equation of the first kind. The next theorem shows that it is the case for
all n = 0, 1, 2, . . . .
Theorem 2.1. The functions uk(x), k = 3, 4, . . . , obtained by the recursion (2.7) with the
initial conditions uk(x), k = 0, 1, given by (2.8) and c0 = 0, c1 = - 1, satisfy the resonant Hermite
differential equation of the first kind.
Proof. We prove this assertion by induction.
Let us assume that up(x), p = 0, 1, . . . , n, all satisfy the resonant Hermite differential equation
of the first kind (2.1). Applying to the recurrence equation (2.7) the Hermite differential operator
\scrA n+1 =
d2
dx2
- 2x
d
dx
+ 2(n+ 1),
and using the induction assumption, we obtain
\scrA n+1un+1(x) = Hn+1(x) +
\biggl[
4
dun(x)
dx
- 8nun - 1(x) + 4Hn - 1(x)
\biggr]
. (2.10)
Further we use the classical relation (see, e.g., [3], \S 10.13)
dHn(x)
dx
= 2nHn - 1(x).
Differentiating this equality by n and using Theorem 3.1 of [4] we get
- 2
dun(x)
dx
= - 4nun - 1(x) + 2Hn - 1(x),
which shows that the square bracket in (2.10) is equal to zero.
Theorem 2.1 is proved.
Remark 2.1. Despite their beauty the formulas (2.9) are uncomfortable for the practical calcu-
lations because it requires differentiation. From this point of view our recurrent algorithm is more
comfortable and can be easily performed using a computer algebra tool like Maple.
Now, the general solution of the resonant equation (2.1) is given by
u(x) = c1Hn(x) + c2hn(x) + un(x), (2.11)
where c1, c2 are arbitrary constants.
2.2. The Hermite resonant equation of the second kind. Let us consider the resonant equation
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
x2
\bigr) d
dx
\biggl[
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- x2
\bigr) dun(x)
dx
\biggr]
+ 2nun(x) = hn(x), (2.12)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. II 459
where hn(x) are the Hermite functions of the second kind defined by (2.4).
Due to Theorem 3.1 of [4] we have a particular solution of the Hermite resonant equation (2.12)
of the second kind in the form
un(x) = ( - 1)[
n+1
2 ]2[
n
2 ]+1
\Bigl(
2
\Bigl[ n
2
\Bigr] \Bigr)
!!
\left[
\biggl(
- n - 1
2
\biggr)
n\biggl(
1
2
\biggr) 2
[n2 ]
x
\partial
\partial \nu
1F1
\biggl(
- \nu
2
+
1
2
;
3
2
;x2
\biggr)
-
-
\Bigl(
- n
2
+ 1
\Bigr)
n - 1\biggl(
1
2
\biggr) 2
[n2 ]
\partial
\partial \nu
1F1
\biggl(
- \nu
2
;
1
2
;x2
\biggr) \right]
\nu =n
, (2.13)
where (a) - 1 = 0. The general solution of (2.12) possesses the form (2.11).
To obtain a recursive algorithm for particular solutions, we differentiate the recurrence equation
for the Hermite functions of the second kind by n and obtain
un+1(x) = 2xun(x) - 2nun - 1(x) + hn - 1(x), n = 1, 2, . . . . (2.14)
From (2.13) we extract the following particular solutions for n = 0, 1:
\chi 0(x) =
\surd
\pi
x\int
0
t\int
0
erfi(\xi ) \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- \xi 2
\bigr)
d\xi \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
t2
\bigr)
dt,
\chi 1(x) =
\left( - 2
x\int
0
\bigl( \surd
\pi erfi(\xi )\xi - \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
\xi 2
\bigr) \bigr) 2
\mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- \xi 2
\bigr)
d\xi x+
(2.15)
+2
x\int
0
\bigl( \surd
\pi erfi(\xi )\xi \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
- \xi 2
\bigr)
- 1
\bigr)
\xi d\xi
\right) \bigl( \surd
\pi erfi(x)x - \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
x2
\bigr) \bigr)
which we modify to the initial values for the recursion (2.14) so that u2(x) satisfies the differential
equation. With this aim we use the ansatzes
u0(x) = \chi 0(x) + c0h0(x), u1(x) = \chi 1(x) + c1h1(x) (2.16)
with undefined coefficients c1, c2. Substituting these into (2.14) with n = 1 we demand that u2(x)
satisfies the resonant differential equation and obtain for the arbitrary constants and for the particular
solution u2(x) the following formulas:
c0 =
3
8
, c1 =
1
4
,
u2(x) = 2xu1(x) - 2u0(x) + h0(x) =
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
460 I. GAVRILYUK, V. MAKAROV
= - 2\chi 0(x) + 2x\chi 1(x) +
\biggl(
x2 +
1
4
\biggr) \surd
\pi erfi(x) - x \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
x2
\bigr)
.
Thus, we have a particular solution in the form
un(x) = - 2Hn - 2(x)\chi 0(x) + pn(x)\chi 1(x) + qn(x)
\surd
\pi erfi(x) + vn(x) \mathrm{e}\mathrm{x}\mathrm{p}
\bigl(
x2
\bigr)
, (2.17)
where the polynomials pn(x) satisfy the recurrence equation for the Hermite polynomials with the
initial conditions p0(x) = 0, p1(x) = 1, the polynomials qn(x) solve the initial value problem
qn+1(x) = 2xqn(x) - 2nqn - 1(x) +Hn - 1(x), n = 1, 2, . . . ,
q0(x) =
3
8
, q1(x) =
x
2
,
and the polynomials vn(x) solve the discrete problem
vn+1(x) = 2xvn(x) - 2nvn - 1(x) + pn - 1(x), n = 1, 2, . . . ,
v0(x) = 0, v1(x) = - 1
2
.
The particular solutions un(x) of the Hermite resonant equation of the second kind satisfy the
resonant differential equation by construction for n = 0, 1, 2. The next theorem shows that it is the
case for all n = 0, 1, 2, . . . .
Theorem 2.2. The functions uk(x) obtained by the recursion (2.17) with the initial conditions
uk(x), k = 0, 1, given by (2.16) satisfy the resonant Hermite differential equation of the second kind
for all k = 3, 4, . . . .
The proof is completely analogous to the one of Theorem 2.1 if we take into account that the
Hermite functions of the second kind (which are not polynomials!) satisfy the same recurrence
equation as the Hermite polynomials and the same differentiation formula.
3. Resonant equation of the Laguerre type. 3.1. The Laguerre resonant equation of the
first kind. In this section we consider the following equation of the Laguerre type:
x
d2u(x)
dx2
+ (1 + \alpha - x)
du(x)
dx
+ nu(x) = L\alpha
n(x), (3.1)
where
L\alpha
n(x) =
(\alpha + 1)n
n!
\Phi ( - n, \alpha + 1, x) =
n\sum
k=0
\Biggl(
n+ \alpha
n - k
\Biggr)
( - x)k
k!
is the Laguerre polynomial satisfying the homogeneous differential equation corresponding to (3.1).
This polynomial can be represented through the confluent hypergeometric function (i.e., through the
solution of a confluent hypergeometric equation, which is a degenerate form of the hypergeometric
differential equation when two of the three regular singularities merge into an irregular singularity)
[1, p. 189] (formula (14)). Since the Laguerre polynomial solves the homogeneous equation, the
inhomogeneous equation (3.1) is resonant.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. II 461
The second linear independent solution of the homogeneous differential equation is the Laguerre
function of the second kind [7, p. 16, 20]. Solving the corresponding differential equation by Maple
we obtain the following representation of the Laguerre function of the second kind for non-integer \alpha :
l\alpha n(x) = x - \alpha
1F1( - n - \alpha ; - \alpha + 1;x) = \Gamma (1 - \alpha , - x)L\alpha
n(x) - ( - x) - \alpha p\alpha n(x) \mathrm{e}\mathrm{x}\mathrm{p}(x),
p\alpha n+1(x) =
1
n+ 1
\bigl[
(2n+ \alpha + 1 - x)p\alpha n(x) - (n+ \alpha )p\alpha n - 1(x)
\bigr]
, n = 1, 2, . . . , (3.2)
p\alpha 0 (x) = 1, p\alpha 1 (x) = 1 - x,
where
\Gamma (a, z) =
\infty \int
z
e - tta - 1dt
is the incomplete Gamma function. For non-negative integer \alpha we have
l\alpha n(x) = Ei1( - x)L\alpha
n(x) - ( - x) - \alpha p\alpha n(x) \mathrm{e}\mathrm{x}\mathrm{p}(x),
p\alpha 1 (x) = - x\alpha +
\alpha \sum
p=1
(p - 1)!(\alpha - p+ 1)+x
\alpha - p, (y)+ =
\left\{ y, y > 0,
0, y \leq 0,
p\alpha 0 (x) = x\alpha - 1 + x\alpha [U(2, 2, - x) + ( - 1)\alpha \alpha !U(1 + \alpha , 1 + \alpha , - x)] =
\alpha \sum
p=1
x\alpha - p(p - 1)!,
where
Ei1(z) =
\infty \int
z
e - t
t
dt, | Arg(z)| < \pi ,
is the exponential integral, and U(a, b, z) is the Kummer’s function of the second kind. The last one
is a solution of the Kummer’s differential equation
z
d2w
dz2
+ (b - z)
dw
dz
- aw = 0.
The other linear independent solution is the Kummer’s function of the first kind M defined, e.g., by
a generalized hypergeometric series:
M(a, b, z) =
\infty \sum
n=0
a(n)z
n
b(n)n!
= 1F1(a; b; z),
where a(0) = 1, a(n) = a(a+ 1)(a+ 2) . . . (a+ n - 1) is the Pochhammer symbol. The Kummer’s
function of the second kind can be represented as
U(a, b, z) =
\Gamma (1 - b)
\Gamma (a+ 1 - b)
M(a, b, z) +
\Gamma (b - 1)
\Gamma (a)
z1 - bM(a+ 1 - b, 2 - b, z).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
462 I. GAVRILYUK, V. MAKAROV
Let f\alpha (z, x) =
\sum \infty
n=0
znp\alpha n(x) be the generating function for the polynomials p\alpha n(x) for both
non-integer and integer non-negative \alpha , then multiplying the the second equation (3.2) by zn and
summing up over n we obtain the Cauchy problem
(1 - z)2
\partial
\partial z
f\alpha (z, x) = [\alpha + 1 - x - z(1 + a)] f\alpha (z, x)+
+
\bigl(
1 + 2z2
\bigr)
p\alpha 1 (x) - (\alpha + 1 - x)p\alpha 0 (x), f\alpha (0, x) = p\alpha 0 (x),
with the solution
f\alpha (z, x) =
\infty \sum
n=0
znp\alpha n(x) = (1 - z) - \alpha - 1 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
x
z - 1
\biggr)
\times
\times
\Biggl\{
-
z\int
0
\bigl(
-
\bigl(
2t2 + 1
\bigr)
p\alpha 1 (x) + (\alpha + 1 - x)p\alpha 0 (x)
\bigr)
(t - 1)\alpha - 1 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
- x
t - 1
\biggr)
dt+
+ ( - 1) - \alpha - 1p\alpha 0 (x) \mathrm{e}\mathrm{x}\mathrm{p}(x)
\Biggr\}
.
In particulary, for \alpha = 0, we have
f0(z, x) = (1 - z) - 1(3 - x) \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
xz
z - 1
\biggr)
+ (1 - z) - 1(x - 1)(x - 3) \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
x
z - 1
\biggr)
\times
\times
\biggl\{
- Ei1( - x) + Ei1
\biggl(
- x
z - 1
\biggr) \biggr\}
+ z - x+ 3,
for \alpha = 1, we obtain
f1(z, x) = (1 - z) - 2
\bigl(
x2 - 5x+ 2
\bigr)
(1 - x)
1
3
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
xz
z - 1
\biggr)
+
+(1 - z) - 2
\bigl(
x3 - 7x2 + 12x - 3
\bigr) x
3
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
x
z - 1
\biggr) \biggl\{
- Ei1( - x) + Ei1
\biggl(
- x
z - 1
\biggr) \biggr\}
-
- 1
3(z - 1)
\bigl[
x3 - (z + 6)x2 +
\bigl(
2z2 + 3z + 7
\bigr)
x - 2z2 - 2z + 1
\bigr]
.
For \alpha = 1/2, it holds
f1/2(z, x) = - (z - 1) - 3/2 \mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
xz
z - 1
\biggr)
\times
\times
\left\{ \mathrm{e}\mathrm{x}\mathrm{p}( - x)
2
z\int
0
\bigl[
1 + 4(x - 1)t2
\bigr]
\surd
t - 1
\mathrm{e}\mathrm{x}\mathrm{p}
\biggl(
x
t - 1
\biggr)
dt+ 1
\right\} .
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RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. II 463
The general solution of the homogeneous Laguerre differential equation is given by
u(x) = c1L
\alpha
n(x) + c2l
\alpha
n(x)
with arbitrary constants c1, c2. Due to Theorem 3.1 of [4] a particular solution of the Laguerre
resonant differential equation of the first kind
x
d2u\nu (x)
dx2
+ (1 - x)
du\nu (x)
dx
+ \nu u\nu (x) = L\nu (x) \equiv \Phi ( - \nu , 1, x)
is given by
u\nu (x) =
d
d\nu
\Phi ( - \nu , 1, x) = -
\infty \sum
k=1
xk
(k!)2
( - \nu )k
p - 1\sum
i=0
1
- \nu + i
,
where \Phi (a, c;x) is the confluent hypergeometric function [1] (Ch. 6). Changing here \nu \in \BbbR to
n \in \BbbN we obtain the resonant Laguerre equation and the corresponding particular solution
un(x) =
n\sum
k=1
xk
(k!)2
d( - \nu )k
d\nu
\bigm| \bigm| \bigm| \bigm|
\nu =n
- ( - 1)nn!
\infty \sum
k=n+1
xk
k!
1\prod n
i=0
(k - i)
= un,1(x) + un,2(x). (3.3)
Using the relation
1\prod n
i=0
(k - i)
=
n\sum
i=0
a
(n)
i
k - n+ i
, a
(n)
i =
( - 1)i
i!(n - i)!
, (3.4)
we transform the sums un,1(x), un,2(x) to a shapes, which can be computed in closed form, e.g., by
Maple and we get
u0(x) = - [Ei1( - x) + \mathrm{l}\mathrm{n}( - x) + \gamma ] ,
u1(x) = - L1(x)u0(x) - \mathrm{e}\mathrm{x}\mathrm{p}(x) + 1 + x,
u2(x) = - L2(x)u0(x) +
1
2
(x - 3) \mathrm{e}\mathrm{x}\mathrm{p}(x) - 3
4
x2 + x+
3
2
,
u3(x) = - L3(x)u0(x) +
1
6
\bigl(
- x2 + 8x - 11
\bigr)
\mathrm{e}\mathrm{x}\mathrm{p}(x) +
11
36
x3 - 7
4
x2 +
1
2
x+
11
6
,
u4(x) = - L4(x)u0(x) +
1
24
\bigl(
x3 - 15x2 + 58x - 50
\bigr)
\mathrm{e}\mathrm{x}\mathrm{p}(x) -
- 25
288
x4 +
19
18
x3 - 7
4
x2 - 1
3
x+
25
12
,
where \gamma = 0.5772156649 . . . is the Euler’s constant. Having in mind to obtain a closed form of the
sum un,2(x), we note that
v
(n)
i (x) =
\infty \sum
p=n+1
xp
p!(p - i)
= - xi
i!
w(x) - \mathrm{e}\mathrm{x}\mathrm{p}(x)
i!
i - 1\sum
p=0
p!xi - 1 - p -
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
464 I. GAVRILYUK, V. MAKAROV
-
n - i - 1\sum
p=0
xn - p
(n - p)!(n - p - i)
+
xi
i!
i - 1\sum
p=0
1
p+ 1
+
i - 1\sum
p=0
xp
p!(i - p)
,
w(x) = Ei1( - x) + \mathrm{l}\mathrm{n}( - x) + \gamma .
(3.5)
Then from (3.3) – (3.5) we have
un,2(x) = - ( - 1)nn!
n\sum
i=0
a
(n)
i v
(n)
n - i(x) =
n\sum
i=0
( - 1)n+1 - iCi
nv
(n)
n - i(x) =
= Ln(x)w(x) - \mathrm{e}\mathrm{x}\mathrm{p}(x)
n - 1\sum
p=0
xp
n - p - 1\sum
i=0
( - 1)n+in!(n - i - 1 - p)!
i![(n - i)!]2
+
+
n\sum
p=0
xp
n\sum
i=0
( - 1)n+1 - in!
i!(n - i)!
bp,i,
where
bp,i =
\left\{
1
p!(i - p)
, p \not = i,
1
i!
\sum i - 1
t=0
1
t+ 1
, p = i.
The technique presented above for \alpha = 0 is even more cumbersome in the case \alpha \not = 0. This is
why below we use our recursive algorithm for the particular solutions in order to be able to write
down the general solution of the Laguerre resonant equation (3.1) in the form
u(x) = c1L
\alpha
n(x) + c2l
\alpha
n(x) + un(x),
with arbitrary constants c1, c2.
Differentiating the recurrence equation for the Laguerre polynomials by n and using Theorem 3.1
of [4] we obtain for the particular solutions the recurrence formula
u\alpha n+1(x) =
2n+ \alpha + 1 - x
n+ 1
u\alpha n(x) -
n+ \alpha
n+ 1
u\alpha n - 1(x)+
+
\alpha - 1 - x
(n+ 1)2
L\alpha
n(x) -
\alpha - 1
(n+ 1)2
L\alpha
n - 1(x), n = 1, 2, . . . , (3.6)
with the corresponding initial conditions. For example, in the case \alpha = 1 we have
u10(x) =
1
x
- \mathrm{l}\mathrm{n}(x), u11(x) = (2 - x)u10(x) - x - 1
x
and the following representation of the particular solution of the resonant equation
un(x) = L1
n(x)u0(x) + qn(x)
\biggl(
x+
1
x
\biggr)
+ vn(x).
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RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. II 465
Here the polynomial qn(x) satisfies the recurrence equation for the Laguerre polynomials with the
initial conditions
v0(x) = 0, v1(x) = - 1.
The polynomial vn(x) solves the difference problem
v1n+1(x) =
2n+ 2 - x
n+ 1
v1n(x) - v1n - 1(x) -
x
(n+ 1)2
L1
n(x), n = 1, 2, . . . ,
v10(x) = 0, v11(x) = 0.
(3.7)
For an arbitrary \alpha due to Theorem 3.1 of [4] we have a particular solution
u\alpha n(x) = - d
d\nu
L\alpha
\nu (x)
\bigm| \bigm| \bigm| \bigm|
\nu =n
= - \Phi ( - n, \alpha + 1;x)
d
d\nu
\Gamma (\alpha + 1 + \nu )
\Gamma (\alpha + 1)\Gamma (\nu + 1)
\bigm| \bigm| \bigm| \bigm|
\nu =n
-
- \Gamma (\alpha + 1 + n)
\Gamma (\alpha + 1)\Gamma (n+ 1)
d
d\nu
\Phi ( - \nu , \alpha + 1;x)
\bigm| \bigm| \bigm| \bigm|
\nu =n
,
from where we obtain the following particular solutions for n = 0, 1:
\chi 0(x) =
x
\alpha + 1
2F2(1, 1; 2, 2 + \alpha ;x),
\chi 1(x) = x 2F2(1, 1; 2, 2 + \alpha ;x) - x2
\alpha + 2
2F2(1, 1; 2, 3 + \alpha ;x).
With the aim to obtain from the recurrence formula solutions of the resonant differential equation we
use the ansatzes
u\alpha 0 (x) = \chi 0(x) + c0, u\alpha 1 (x) = \chi 1(x) + c1L
\alpha
1 (x)
with undefined coefficients c0, c1. Substituting these into (3.7) and demanding, that the particular
solution u\alpha 1 (x) satisfies the resonant differential equation we get
c0 = - \alpha (3\alpha + 5)
2(\alpha + 1)(\alpha + 2)
, c1 = - \alpha
2(\alpha + 2)
.
Now, the initial values for the recursive algorithm for the particular solutions become to
u\alpha 0 (x) =
x
\alpha + 1
2F2(1, 1; 2, 2 + \alpha ;x) - \alpha (3\alpha + 5)
2(\alpha + 1)(\alpha + 2)
,
u\alpha 1 (x) = x 2F2(1, 1; 2, 2 + \alpha ;x) - x2
\alpha + 2
2F2(1, 1; 2, 3 + \alpha ;x) - \alpha
2(\alpha + 2)
L\alpha
1 (x).
(3.8)
The next assertion shows that the functions u\alpha n(x) generated by recursion (3.6) with the initial
values (3.8) satisfy the Laguerre resonant differential equation of the first kind for all n = 0, 1, 2, . . . .
Theorem 3.1. The functions u\alpha n(x) generated by the recursive algorithm (3.6) with the initial
values (3.8) are particular solutions of the Laguerre resonant differential equation of the first kind
for all n = 0, 1, 2, . . . .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
466 I. GAVRILYUK, V. MAKAROV
Proof. We prove the assertion by the mathematical induction. First of all we note that the
functions u\alpha n(x) for n = 0, 1, 2 are the particular solution due to their construction. We assume
that all functions u\alpha p (x), p = 0, 1, . . . , n, are particular solutions and prove that then u\alpha n+1(x) is a
solution too.
Actually, the application to the both sides of (3.6) of the Laguerre differential operator
\scrA \alpha
n+1 = x
d2
dx2
+ (\alpha + 1 - x)
d
dx
+ n+ 1
and the induction assumption provide
\scrA \alpha
n+1u
\alpha
n+1(x) = L\alpha
n+1(x) +
2
n+ 1
\biggl[
nu\alpha n(x) - x
du\alpha n(x)
dx
- (n+ \alpha )u\alpha n - 1(x)
\biggr]
-
- 2
n+ 1
\bigl[
L\alpha
n(x) - L\alpha
n - 1(x)
\bigr]
. (3.9)
Further we use the relation (see, e.g., [8], \S 10.12)
x
dL\alpha
n(x)
dx
= nL\alpha
n(x) - (n+ \alpha )L\alpha
n - 1(x).
Differentiating this relation by n and using Theorem 3.1 of [4] we see that the both square brackets
in (3.9) are equal to zero and herewith the assertion is proven.
The general representation of the particular solutions is
u\alpha n(x) = p\alpha n(x) 2F2(1, 1; 2, 2 + \alpha ;x) + q\alpha n(x) 2F2(1, 1; 2, 3 + \alpha ;x) + v\alpha n(x), n = 2, 3, . . . ,
where the polynomials p\alpha n(x), q
\alpha
n(x) satisfy the classical Laguerre recurrence equation with the initial
conditions
p\alpha 0 (x) =
x
\alpha + 1
, p\alpha 1 (x) = x,
q\alpha 0 (x) = 0, q\alpha 1 (x) = - x2
\alpha + 2
,
respectively. The polynomials v\alpha n(x) satisfies the inhomogeneous recurrence equation
v\alpha n+1(x) =
2n+ \alpha + 1 - x
n+ 1
v\alpha n(x) -
n+ \alpha
n+ 1
v\alpha n - 1(x)+
+
\alpha - 1 - x
(n+ 1)2
L\alpha
n(x) -
\alpha - 1
(n+ 1)2
L\alpha
n - 1(x), n = 1, 2, . . . ,
with the initial conditions
v\alpha 0 (x) = - \alpha (3\alpha + 5)
2(\alpha + 1)(\alpha + 2)
, v\alpha 1 (x) = - \alpha (\alpha + 1 - x)
2(\alpha + 2)
.
3.2. The Laguerre resonant equation of the first kind (revisited). In this section we consider
again the resonant Laguerre differential equation of the first type (3.1) and show that the particular
solutions can be represented by elementary functions only.
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RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. II 467
We know that one of the linear independent solutions of the homogeneous differential equation
is the Laguerre function of the second kind [7, p. 16, 20]. Solving the corresponding differential
equation by Maple we obtain the following representation of the Laguerre function of the second
kind for non-integer \alpha :
l\alpha n(x) = x - \alpha
1F1( - n - \alpha , - \alpha + 1;x) = \Gamma (1 - \alpha , - x)L\alpha
n(x) - ( - x) - \alpha p\alpha n(x) \mathrm{e}\mathrm{x}\mathrm{p}(x),
p\alpha n+1(x) =
1
n+ 1
\bigl[
(2n+ \alpha + 1 - x)p\alpha n(x) - (n+ \alpha )p\alpha n - 1(x)
\bigr]
, n = 1, 2, . . . ,
p\alpha 0 (x) = 0, p\alpha 1 (x) = 1 - x.
For non-negative natural \alpha \in \BbbN we have
l\alpha n(x) = Ei1( - x)L\alpha
n(x) - ( - x) - \alpha p\alpha n(x) \mathrm{e}\mathrm{x}\mathrm{p}(x),
p\alpha - 1(x) = (\alpha - 1)!, (3.10)
p\alpha 0 (x) = x\alpha - 1 + x\alpha [U(2, 2, - x) + ( - 1)\alpha \alpha !U(1 + \alpha , 1 + \alpha , - x)] .
Note that the function at the second initial condition in (3.10) solves the following difference
initial value problem:
p\alpha 0 (x) = xp\alpha - 1
0 (x) + (\alpha - 1)!, \alpha = 1, 2, . . . , p00(x) = 0.
Using Theorem 3.1 of [4] we can represent the particular solutions of the Laguerre resonant
equation of the first kind by
un(x) =
( - 1)n+1
n!
\partial
\partial \nu
U( - \nu , 1 + \alpha , - x)| n=\nu , n = 0, 1 . . . .
This representation provides the particular solutions
\chi \alpha
0 (x) = u0(x) = - \mathrm{l}\mathrm{n}(x) +
\alpha - 1\sum
p=0
(\alpha - p)p+1
(p+ 1)xp+1
,
\chi \alpha
1 (x) = u1(x) = - L\alpha
1 (x) \mathrm{l}\mathrm{n}(x) +
\alpha \sum
p=0
kp(\alpha )
xp
,
where
kp+1(\alpha ) = p
\alpha - 1\sum
i=1
kp(i), p = 1, 2, . . . , \alpha - 1,
k1(\alpha ) =
\alpha (\alpha + 1)
2
, k0(\alpha ) = - \alpha - 2, \alpha = 2, 3, . . . .
At the first step of Algorithm 3.1 of [4] we use the ansatzes
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
468 I. GAVRILYUK, V. MAKAROV
u\alpha 0 (x) = \chi \alpha
0 (x) + c0L
\alpha
0 (x) + d0L
\alpha
1 (x),
u\alpha 1 (x) = \chi \alpha
1 (x) + c1L
\alpha
0 (x) + d1L
\alpha
1 (x)
with undefined coefficients c0, d0, c1, d1, substitute them into (3.6) with n = 1, obtain u\alpha 2 (x) and
choose c0, d0, c1, d1 so that u\alpha 2 (x) satisfies the resonant differential equation. We get d0 = 0,
d1 = 0 and c1 = 1 + c0.
Now one can prove that
u\alpha n(x) = - L\alpha
n(x) \mathrm{l}\mathrm{n}(x) +
p\alpha n(x)
x\alpha
,
where the polynomials p\alpha n(x) satisfy the recurrence equation
p\alpha n+1(x) =
2n+ \alpha + 1 - x
n+ 1
p\alpha n(x) -
n+ \alpha
n+ 1
p\alpha n - 1(x)+
+
\alpha - 1 - x
(n+ 1)2
L\alpha
n(x) -
\alpha - 1
(n+ 1)2
L\alpha
n - 1(x), n = 1, 2, . . . ,
with the initial conditions
p\alpha 0 (x) =
\alpha - 1\sum
p=0
x\alpha - p - 1(\alpha - p)p+1
p+ 1
+ c0x
\alpha , p\alpha 1 (x) =
\alpha \sum
p=0
x\alpha - pkp(\alpha ) + (1 + c0)x
\alpha L\alpha
1 (x).
3.3. The Laguerre resonant equation of the second kind. In this subsection we consider the
resonant equation
x
d2u(x)
dx2
+ (1 + \alpha - x)
du(x)
dx
+ nu(x) = l\alpha n(x), (3.11)
where l\alpha n(x) is the Laguerre function of the second kind given by (3.2).
Due to Theorem 3.1 of [4] the formula
un(x) = - d
d\nu
l\alpha \nu (x)| \nu =n (3.12)
defines a particular solution of (3.11), so that its general solution is given by
u(x) = c1L
\alpha
n(x) + c2l
\alpha
n(x) + un(x).
The use of formula (3.12) for arbitrary n is rather burdensome, therefore we use Algorithm 3.1
of [4], where we for the sake of simplicity set \alpha = 0. Solving differential equation (3.11) with Maple
for n = 0, n = 1 we get
\chi 0(x) = -
x\int
1
\mathrm{e}\mathrm{x}\mathrm{p}(t)
t
t\int
1
Ei1( - \xi ) \mathrm{e}\mathrm{x}\mathrm{p}( - \xi )d\xi dt,
(3.13)
\chi 1(x) = [(1 - x)Ei1( - x) - \mathrm{e}\mathrm{x}\mathrm{p}(x)]
x\int
1
[1 + Ei1( - \xi )( - 1 + \xi ) \mathrm{e}\mathrm{x}\mathrm{p}( - \xi )] ( - 1 + \xi )d\xi +
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RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. II 469
+
x\int
1
\mathrm{e}\mathrm{x}\mathrm{p}( - \xi ) [Ei1( - \xi )( - 1 + \xi ) + \mathrm{e}\mathrm{x}\mathrm{p}( - \xi )]2 d\xi ( - 1 + x).
As the ansatzes for initial values of our algorithm we use
u00(x) = \chi 0(x) + c0Ei1( - x) + d0, u01(x) = \chi 1(x) + c1l
0
1(x) + d1L
0
1(x) (3.14)
with undefined constants c0, d0, c1, d1. Differentiating the recurrence equation for the Laguerre
functions of the second kind by n in regard of (3.12) we obtain the following recurrence relation for
particular solutions:
u0n+1(x) =
2n+ 1 - x
n+ 1
u0n(x) -
n
n+ 1
u0n - 1(x) -
1 + x
(n+ 1)2
l0n(x) +
1
(n+ 1)2
l0n - 1(x). (3.15)
We substitute (3.14) into this equation with n = 1 and demand that the obtained function u02(x)
satisfies the resonant differential equation (3.11) with n = 2, then we obtain
c0 = - Ei1( - 1) \mathrm{e}\mathrm{x}\mathrm{p}( - 1) - 1, d0 = - [Ei1( - 1) \mathrm{e}\mathrm{x}\mathrm{p}( - 1/2) + \mathrm{e}\mathrm{x}\mathrm{p}(1/2)]2,
c1 = 0, d1 = 0.
(3.16)
Analogously to Theorem 3.1 the following assertion can be proven.
Theorem 3.2. The functions u0n(x) generated by the recursive algorithm (3.15) with the initial
values (3.14) with the constants given by (3.16) are particular solutions of the Laguerre resonant
differential equation of the second kind for all n = 0, 1, 2, . . . .
It can be proven by substitution into (3.15) that the following representation holds true:
u0n(x) = p0n(x)\chi 1(x) + q0n(x)\chi 0(x) + v0n(x)Ei1( - x) + w0
n(x) \mathrm{e}\mathrm{x}\mathrm{p}(x) + q0n(x)d0, (3.17)
where the polynomials p0n(x), q0n(x) satisfy the recurrence relation for the Laguerre polynomials
with the initial conditions
p00(x) = 0, p01(x) = 1, q00(x) = 1, q01(x) = 0.
The polynomials w0
n(x) satisfy the inhomogeneous recurrence relation for the Laguerre polynomials
w0
n+1(x) =
2n+ 1 - x
n+ 1
w0
n(x) -
n
n+ 1
w0
n - 1(x) -
- 1 + x
(n+ 1)2
p0n(x) +
1
(n+ 1)2
p0n - 1(x), n = 1, 2, . . . ,
with the initial conditions
w0
1(x) = 0, w0
2(x) =
x+ 1
4
.
Here p0n(x) are the same polynomials as in (3.17).
The polynomials v0n(x) solve the following discrete initial value problem:
v0n+1(x) =
2n+ 1 - x
n+ 1
v0n(x) -
n
n+ 1
v0n - 1(x) -
1 + x
(n+ 1)2
L0
n(x)+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
470 I. GAVRILYUK, V. MAKAROV
+
1
(n+ 1)2
L0
n - 1(x), n = 1, 2, . . . ,
v01(x) = 0, v02(x) =
x2 - 2c0
4
.
Below we give some particular solutions of the Laguerre resonant equation of the second kind
obtained by our algorithm:
u00(x) = \chi 0(x) + c0Ei1( - x) + d0, u01(x) = \chi 1(x),
u02(x) = - x - 3
2
\chi 1(x) -
1
2
\chi 0(x) +
x2 - 2c0
4
Ei1( - x) - x2 - 1
8
\mathrm{e}\mathrm{x}\mathrm{p}(x) - 1
2
d0,
u03(x) =
\biggl(
1
6
x2 - 4
3
x+
11
6
\biggr)
\chi 1(x) +
\biggl(
1
6
x - 5
6
\biggr)
\chi 0(x)+
+
\biggl(
- 5
36
x3 +
7
12
x2 +
c0
6
x - 5c0
6
\biggr)
Ei1( - x)+
+
\biggl(
1
24
x3 - 11
72
x2 - 23
72
x - 1
72
\biggr)
\mathrm{e}\mathrm{x}\mathrm{p}(x) +
\biggl(
1
6
x - 5
6
\biggr)
d0,
where c0, d0 are given by (3.16) and \chi 0, \chi 1 — by (3.3).
References
1. Bateman H., Erdélyi A. Higher trancendental functions. – New York etc.: McGraw-Hill Book Co., Inc., 1953.
2. Bateman H., Erdélyi A. Higher trancendental functions. – New York etc.: McGraw-Hill Book Co., Inc., 1953.
3. Backhouse N. B. The resonant Legendre equation // J. Math. Anal. and Appl. – 1986. – 133.
4. Gavrilyuk I., Makarov V. Resonant equations with classical orthogonal polynomials. I // Ukr. Mat. Zh. – 2019. – 71,
№ 2. – P. 190 – 209.
5. Krazer A., Franz W. Transzendente Funktionen. – Akademie Verlag, 1960.
6. Nikiforov F., Uvarov V. Special functions of the mathematical physics. – Moscow: Nauka, 1978 (in Russian).
7. Parke W. C., Maximon L. C. On second solutions to second-order difference equations // arXiv:1601.04412 [math.CA]
Received 21.11.18
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
|
| id | umjimathkievua-article-1451 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:40Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/24/a371618dff88251ba229d9e14d2f7524.pdf |
| spelling | umjimathkievua-article-14512019-12-05T10:12:29Z Resonant equations with classical orthogonal polynomials. II Резонанснi рiвняння з класичними ортогональними полiномами.II Gavrilyuk, I. P. Makarov, V. L. Гаврилюк, І. П. Макаров, В. Л. UDC 517.9 We study some resonant equations related to the classical orthogonal polynomials on infinite intervals, i.e., the Hermite and the Laguerre orthogonal polynomials, and propose an algorithm of finding their particular and general solutions in the closed form. The algorithm is especially suitable for the computer-algebra tools, such as Maple. The resonant equations form an essential part of various applications, e.g., of the efficient functional-discrete method for the solution of operator equations and eigenvalue problems. These equations also appear in the context of supersymmetric Casimir operators for the di-spin algebra, as well as of the square operator equations $A^2u = f$ , e.g., of the biharmonic equation. УДК 517.9 Вивчаються резонанснi рiвняння, що пов’язанi з класичними ортогональними многочленами, заданими на нескiнченних iнтервалах, тобто з ортогональними многочленами Ермiта i Лагерра. Запропоновано алгоритм знаходження їхнiх частинних розв’язкiв i загального розв’язку в замкненому виглядi. Цей алгоритм є особливо зручним в iмплементацiї засобами комп’ютерної алгебри, наприклад, Maple. Резонанснi рiвняння є вагомою складовою рiзних застосувань, наприклад ефективного функцiонально-дискретного методу розв’язування операторних рiвнянь i задач на власнi значення. Такi рiвняння виникають також у контекстi суперсиметричних операторiв Казимiра для дi-спiнової алгебри, а також при розв’язуваннi операторних рiвнянь з квадратом деякого оператора, наприклад бiгармонiчного рiвняння. Institute of Mathematics, NAS of Ukraine 2019-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1451 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 4 (2019); 455-470 Український математичний журнал; Том 71 № 4 (2019); 455-470 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1451/435 Copyright (c) 2019 Gavrilyuk I. P.; Makarov V. L. |
| spellingShingle | Gavrilyuk, I. P. Makarov, V. L. Гаврилюк, І. П. Макаров, В. Л. Resonant equations with classical orthogonal polynomials. II |
| title | Resonant equations with classical orthogonal polynomials. II |
| title_alt | Резонанснi рiвняння з класичними ортогональними полiномами.II |
| title_full | Resonant equations with classical orthogonal polynomials. II |
| title_fullStr | Resonant equations with classical orthogonal polynomials. II |
| title_full_unstemmed | Resonant equations with classical orthogonal polynomials. II |
| title_short | Resonant equations with classical orthogonal polynomials. II |
| title_sort | resonant equations with classical orthogonal polynomials. ii |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1451 |
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