Resonant equations with classical orthogonal polynomials. I
In the present paper, we study some resonant equations related to the classical orthogonal polynomials and propose an algorithm of finding their particular and general solutions in the explicit form. The algorithm is especially suitable for the computer algebra tools, such as Maple. The resonant equ...
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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. |
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| description | In the present paper, we study some resonant equations related to the classical orthogonal polynomials and propose an
algorithm of finding their particular and general solutions in the explicit 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 aimed at 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 for the square operator
equations $A^2u = f$; e.g., for the biharmonic equation. |
| first_indexed | 2026-03-24T02:05:13Z |
| 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. I
РЕЗОНАНСНI РIВНЯННЯ З КЛАСИЧНИМИ
ОРТОГОНАЛЬНИМИ ПОЛIНОМАМИ. I
In the present paper, we study some resonant equations related to the classical orthogonal polynomials and propose an
algorithm of finding their particular and general solutions in the explicit 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 aimed at 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 for the square operator
equations A2u = f ; e.g., for the biharmonic equation.
Вивчаються деякi резонанснi рiвняння, що мають вiдношення до класичних ортогональних полiномiв. Запропо-
новано алгоритм знаходження їхнiх частинних та загальних розв’язкiв у явному виглядi. Цей алгоритм найкрaще
пiдходить для методiв комп’ютерної алгебри, таких як Maple. Резонанснi рiвняння складають суттєву частину ба-
гатьох застосувань, зокрема ефективного функцiонально-дискретного методу, що застосовується при розв’язаннi
операторних рiвнянь та задач на власнi значення. Такi рiвняння також з’являються в контекстi суперсиметричних
операторiв Казимiра для дi-спiнової алгебри, а також для квадратичних операторних рiвнянь A2u = f, наприклад
для бiгармонiчного рiвняння.
1. Introduction. Polynomials, especially the orthogonal polynomials [8, 9, 26] is a very important
and often used mathematical tool. One of the application fields for polynomials are differential equa-
tions. Some of them possess polynomial solutions and the solution of other ones can be approximated
by polynomials. The topic of the present paper is a special class of resonant differential equations
with differential operators related to the classical orthogonal polynomials.
There are various definitions of resonant equations (see, e.g., [1, 2]), where a boundary-value
problem is called resonant, when the operator, defined by the differential equation and by the boun-
dary conditions does not possess the inverse. In the present paper we follow the definition from
[6, 17, 19] and call an equation of the form Lf = g with Lg = 0 resonant. In other words, the right-
hand side of the resonant equation belongs to the kernel K(L) of the operator L. These equations
are interesting both from theoretical point of view and from the practical side in various applications.
For example, in [18] was proposed the so-called functional-discrete method (FD-method) for solving
of operator equations and of eigenvalue problems. The method is based on the ideas of perturbation
of the operator involved and on the homotopy idea. This approach was applied to various problems
in particulary to eigenvalue problems in [10 – 14] and has been proven to possess a super exponential
convergence rate. An essential part of the algorithm are some inhomogeneous equations with a
resonant component in the sense of the definition above.
A simple but profound example showing the principally different behaviors of the solutions in
the resonant and in the non-resonant cases gives the following simple differential equation (so-called
vibration equation):
d2y
dt2
+ \mu 2y = \mathrm{s}\mathrm{i}\mathrm{n} (\nu t). (1.1)
c\bigcirc I. GAVRILYUK, V. MAKAROV, 2019
190 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. I 191
There exists a particular solution of the form
y(t) =
\left\{
1
\mu 2 - \nu 2
\mathrm{s}\mathrm{i}\mathrm{n} (\nu t), if \nu \not = \pm \mu ,
- t
2\mu
\mathrm{c}\mathrm{o}\mathrm{s} (\mu t), if \nu = \pm \mu ,
(1.2)
which is resonant for \nu = \pm \mu
\Bigl(
in this case the right-hand side \mathrm{s}\mathrm{i}\mathrm{n} (\mu t) solves the homogeneous
equation
d2y
dt2
+ \mu 2y = 0
\Bigr)
and non-resonant otherwise. The vibration amplitude in the resonant case
tends to infinity if the stimulating vibration frequency \nu on the right-hand side of the equation tends
to the resonant eigenfrequency \mu of the system described by the differential operator on the left-hand
side. The depth of difference between the non-resonant and the resonant solutions is discussed in [7].
The example above can be embedded into the following abstract framework. Let some system be
described by an operator equation Au - \lambda u = f in some Hilbert space H, where the operator A is
completely defined by its spectrum, i.e., by the eigenvalues \lambda j , j = 1, 2, . . . , and the corresponding
eigenvectors uj , j = 1, 2, . . . . Here \lambda is a parameter characterizing the system. If the right-hand
side is of the form f = \alpha uk for the fixed \alpha , k, i.e., f solves the equation (A - \lambda k)f = 0, then the
solution of the corresponding operator equation is
u =
\alpha
\lambda k - \lambda
uk.
We have \| u\| \rightarrow \infty (\| u\| can be interpreted as the amplitude in the example above) in two cases:
1) if \alpha \rightarrow \infty (the stimulating amplitude tends to infinity) and 2) if the system parameter tends to an
eigenvalue \lambda k of the operator, i.e., \lambda \rightarrow \lambda k. The second case is called resonance and the value \lambda k of
the parameter \lambda is called the resonant value. In this case we deal with the resonant equation in the
sense of our definition above and of the example equation (1.1). It is clear, that a system can possess
various resonant parameter values.
The resonant equations arise also when solving the quadratic operator equation
A2u = 0 (1.3)
with a given operator A. Denoting Au = v we reduce equation (1.3) to the “simpler” pair of
equations Av = 0, Au = v where the last equation is resonant.
The resonant phenomena play a very important role in the natural world, in various technical
applications, e.g. magnetic resonance imaging (nuclear spin tomography) [24], fluid dynamics [15,
16] etc. The resonant equations arose also in the context of supersymmetric Casimir operators for the
di-spin algebra (see, e.g., [6, 7] and the literature cited therein). These equations often require specific
solution and investigation techniques [1 – 3]. The solvability condition of a resonant equation in a
Hilbert space is the orthogonality of the right-hand side to the kernel of the operator. The situation for
other spaces is more complicated (see, e.g., [1 – 3]), where some sufficient solvability conditions has
been proven. For example, in [2] the following Liénard equation, describing vibrations and various
dynamical systems, was considered as an illustration of the presented solvability theory:
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
192 I. GAVRILYUK, V. MAKAROV
\"x(t) + g(x) \.x(t) + ax(t) = f(t), x(0) = x(1), \.x(0) = \.x(1), t \in [0, 1]. (1.4)
Here a is a constant, the function g : R1 \rightarrow R1 is supposed to be continuous and the solution is
looking for in the class of twice continuously differentiable on [0, 1] functions. It was shown that
this problem possesses at least one solution for arbitrary function
f(t) :
1\int
0
f(t)dt = 0
provided that
| g(x)| \leq b, a \in R1, b+ 4| a| /3 < 1. (1.5)
The simple counterexample with f(t) =
4
3\pi
\mathrm{c}\mathrm{o}\mathrm{s}(2\pi t) + \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t), g(x) \equiv 0, a = 4\pi 2 (here the first
summand is a resonant component) shows that the differential equation is resonant in the sense of
our definition, conditions (1.5) due to b = 0, a = 4\pi 2 as well as the condition
\int 1
0
f(t)dt = 0 are
not fulfilled, but there exists a set of solutions given by
u(t) = C \mathrm{c}\mathrm{o}\mathrm{s}(2\pi t) +D \mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) +
t
3\pi 2
\mathrm{s}\mathrm{i}\mathrm{n}(2\pi t) +
1
3\pi 2
\mathrm{s}\mathrm{i}\mathrm{n}(\pi t) \forall C,D \in R1,
i.e., conditions (1.5) are rather coarse.
In the present paper we consider resonant equations with the differential operators of the hyperge-
ometric type which define the classical orthogonal polynomials. The solutions of such homogeneous
differential equation is the corresponding orthogonal polynomial (or the solution of the first kind)
and the second linear independent solution is the so-called function of the second kind, so that the
general solution is a linear combination of both. The inhomogeneous differential equations with
the corresponding orthogonal polynomial or the function of the second kind on the right-hand side
are resonant equations correspondingly of the first and of the second kind. We need their particular
solutions to write down the general solution of the inhomogeneous resonant equation.
We propose a general algorithm to find such particular solutions explicitly, therefore, we can
obtain the general solutions of the inhomogeneous resonant equations of the first and of the second
kind in explicit form. This algorithm is especially suitable for the computer algebra tools like Maple
etc. Besides, it provides a constructive proof of the existence of the solutions too.
The paper consists of two parts and is organized as follows. In Section 2 we show that the
resonant equations are a natural part of the FD-method. The main result of the Section 3 is Theorem
3.1, giving a formula for particular solutions of a resonant operator equation depending on some
parameter. This section contains also the description of the general Algorithm 3.1 to compute the
particular solutions of the inhomogeneous resonant equations with the differential operators related
to the classical orthogonal polynomials. Theorem 3.1 plays the crucial role for the justification of our
algorithm. Each of the next two sections consists of two subsections devoted to the corresponding
resonant equations of the first and of the second kind with the differential operators related to the
classical orthogonal polynomials of Legendre and Jacobi types. The explicit formulas for the general
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. I 193
solutions of the corresponding inhomogeneous resonant differential equations are given. The clas-
sical orthogonal polynomials defined on the infinite intervals, namely the Hermite and the Laguerre
polynomials are the topics of part II. With the aim to emphasize the advantages of our algorithm we
give the particular solutions through the hypergeometric or confluent hypergeometric functions too.
2. The homotopy based method for the eigenvalue problems. Let us briefly explain the ideas
of perturbation and homotopy for the eigenvalue problem
(A+B)un - \lambda nun = \theta (2.1)
in a Hilbert space X with a scalar product (\cdot , \cdot ) and with the null-element \theta under the assumption
that the spectrum of the operator A+B is discrete and we are looking for the eigenpair with a given
fixed index n.
Let B be an approximating operator for B in the sense that the eigenvalue problem
(A+B)u(0)n - \lambda (0)n u(0)n = \theta (2.2)
is “simpler” then problem (2.1).
Formally, a homotopy between two problems P1 and P2 with solutions u1 and u2 from some
topological space X is defined to be a parametric problem PH(t) with a solution u(t) continuously
depending on the parameter t \in [0, 1] and such that u(0) = u1 and u(1) = u2 (compare with
http://en.wikipedia.org/wiki/Homotopy).
Following to the homotopy idea for a given eigenpair number n we imbed our problem into the
parametric family of problems
(A+W (t))un(t) - \lambda n(t)un(t) = \theta , t \in [0, 1], (2.3)
with W (t) = B + t\varphi (B), \varphi (B) = B - B, where B is some approximation of B. This family
contains the both problems (2.1) and (2.2), so that we obviously have
un(0) = u(0)n , \lambda n(0) = \lambda (0)n , un(1) = un, \lambda n(1) = \lambda n. (2.4)
This suggests the idea to look for the solution of (2.3) as the Taylor series
\lambda n(t) =
\infty \sum
j=0
\lambda (j)n tj , un(t) =
\infty \sum
j=0
u(j)n tj , (2.5)
where formally
\lambda (j)n =
1
j!
dj\lambda n(t)
dtj
\bigm| \bigm| \bigm| \bigm|
t=0
, u(j)n =
1
j!
djun(t)
dtj
\bigm| \bigm| \bigm| \bigm|
t=0
. (2.6)
Setting t = 1 in (2.5) we obtain
\lambda n =
\infty \sum
j=0
\lambda (j)n , un =
\infty \sum
j=0
u(j)n (2.7)
provided that series (2.5) converge for all t \in [0, 1]. The truncated series
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
194 I. GAVRILYUK, V. MAKAROV
m
\lambda n =
\infty \sum
j=0
\lambda (j)n ,
m
un =
\infty \sum
j=0
u(j)n (2.8)
represent a computational algorithm of rank m.
The formulas (2.6) are not suitable for a numerical algorithm, therefore we need an other way to
compute the corrections \lambda (j)n , u
(j)
n which we describe below.
Substituting (2.5) into (2.3) and matching the coefficients in front of the same powers of t we
arrive at the following recurrence sequence of equations:
(A+B)u(j+1)
n - \lambda (0)n u(j+1)
n = F (j+1)
n , j = - 1, 0, 1, . . . , (2.9)
with F (0)
n = 0 and
F (j+1)
n = F (j+1)
n (\lambda (0)n , . . . , \lambda (j+1)
n ;u(0)n , . . . , u(j)n ) = - \varphi (B)u(j)n +
j\sum
p=0
\lambda (j+1 - p)
n u(p)n =
= \lambda (j+1)
n u(0)n - \varphi (B)u(j)n +
j\sum
p=1
\lambda (j+1 - p)
n u(p)n , j = - 1, 0, 1, . . . . (2.10)
For the pair \lambda (0)n , u
(0)
n corresponding to the index j = - 1 we have the so-called base eigenvalue
problem
(A+B)u(0)n - \lambda (0)n u(0)n = \theta (2.11)
in a Hilbert space which is assumed to have not multiple eigenvalues and to be “simpler” then
the original one and produces the initial data for problems (2.9), (2.10). We suppose that u(0)n ,
n = 1, 2, . . . , is a basis of the corresponding Hilbert space. The case of the base problems with
multiple eigenvalues was studied in [10, 20, 21].
Each problem (2.10) contains in the right-hand side the summand \lambda (j+1)
n u
(0)
n which solves the ho-
mogeneous equation with the same operator, i.e., the solution u(j+1)
n of (2.10) contains a component
which is the solution of the corresponding resonant equation.
Problems (2.9) for higher indices j \geq 0 are solvable provided that
(F (j+1)
n , u(0)n ) = 0, j = 0, 1, . . . . (2.12)
Supposing additionally (for uniqueness)
(u(j+1)
n , u(0)n ) = 0, j = 0, 1, . . . , (2.13)
we obtain
\lambda (j+1)
n = (\varphi (B)u(j)n , u(0)n ), j = 0, 1, . . . . (2.14)
Under these conditions we obtain the particular solution
u(j+1)
n =
\infty \sum
p=1,p \not =n
((F
(j)
n , u
(0)
p ))
\lambda
(0)
p - \lambda
(0)
n
u(0)p (2.15)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. I 195
satisfying condition (2.13). The start values \lambda (0)n , u
(0)
n for the recursion (2.9), (2.14) is the solution
of the base problem.
The next theorem [18] gives the error estimates of the method above and its convergence as
m\rightarrow \infty .
Theorem 2.1. Let A be a closed operator in a Hilbert space H, problem (2.11) possesses a
discrete spectrum of eigenvalues 0 \leq \lambda
(0)
1 < \lambda
(0)
2 < . . . and the corresponding eigenvectors u(0)n ,
n = 1, 2, . . . , represent a basis of H. Let the inequality
qn = 4Mn\| \varphi (B)\| < 1 (2.16)
with
Mn = \mathrm{m}\mathrm{a}\mathrm{x}
\Biggl\{
1
\lambda
(0)
n - \lambda
(0)
n - 1
,
1
\lambda
(0)
n+1 - \lambda
(0)
n
\Biggr\}
(2.17)
holds true. Then the series (2.7) converge to the solution \lambda n, un of problem (2.1) and the accuracy
of algorithm (2.8) is given by the estimates
\| un - m
un\| \leq \alpha m+1
qm+1
n
1 - qn
,
\| \lambda n -
m
\lambda n\| \leq \| \varphi (B)\| \alpha m
qmn
1 - qn
,
(2.18)
where
\alpha m = 2
(2m - 1)!!
(2m+ 21)!!
. (2.19)
3. Representation of particular solutions. This section deals with particular solutions of the
resonant equations. We give a representation of particular solutions of a resonant equation in a Banach
space. Besides we propose an algorithm to compute particular solutions of resonant equations with
the differential operators related to the classical orthogonal polynomials.
The following result has been proven in [19].
Theorem 3.1. Let A : X \rightarrow X be a linear operator acting in a Banach space X, the set
K(A) \subset X be the kernel of A and a connected set \Sigma (A) in the complex plane be the spectral set of
A. If f(\lambda ) \in K(A - \lambda E), \lambda \in \Sigma (A) is a differentiable function, then the solution of the resonant
equation
(A - \lambda E)u = f(\lambda ) (3.1)
can be represented by
u(\lambda ) =
df(\lambda )
d\lambda
. (3.2)
The proof of this theorem is based on the equivalent equation (A - \lambda 0E)
f(\lambda ) - f(\lambda 0)
\lambda - \lambda 0
= f(\lambda )
with some fixed \lambda 0 and on passing to the limit \lambda \rightarrow \lambda 0.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
196 I. GAVRILYUK, V. MAKAROV
Now, let
\scrA n = \sigma (x)
d 2
dx2
+ \tau (x)
d
dx
+ \lambda n (3.3)
be a differential operator of the hypergeometric type with a polynomial \sigma (x) of the degree not greater
then two, a polynomial \tau (x) of the degree not greater then one and a constant \lambda n and Pn(x) be
some classical orthogonal polynomial satisfying the homogeneous differential equation
\scrA nPn(x) = 0 (3.4)
(see, e.g., [5, 23, 25]). We call the polynomial solution Pn(x) of this homogeneous differential
equation the function of the first kind. Let Qn(x) be the second linear independent solution of the
homogeneous differential equation, which is called the function of the second kind.
Let us consider the resonant equations of the type
\scrA nun(x) = Rn(x). (3.5)
In the case when Rn(x) is the classical orthogonal polynomial Pn(x) (the function of the first kind),
the inhomogeneous differential equation (3.5) is called the resonant equation of the first kind. The
inhomogeneous differential equation of the type (3.5) with the right-hand side Qn(x) instead of
Rn(x) is called the resonant differential equation of the second kind. Both functions Pn(x) and
Qn(x) satisfy the same homogeneous differential equation (3.4) and the same recurrence equation
Rn+1(x) = (\alpha nx+ \beta n)Rn(x) - \gamma nRn - 1(x), n = 1, 2, . . . , (3.6)
with some constants \alpha n, \beta n, \gamma n (see, e.g., [5, 22, 23, 25]). Since our algorithm below for particular
solutions of the resonant differential equations of the first and of the second kind (3.5) is based on
the same recurrence relation (3.6) it is valid for the resonant equations of both types and we use the
notation Rn(x) below for both Pn(x) and Qn(x).
Algorithm 3.1. 1. Using Theorem 3.1 we find some particular solutions of (3.5) for n = 0, 1,
i.e.,
\chi 0(x) = - 1
\lambda \prime (\nu )
dR\nu (x)
d\nu
\bigm| \bigm| \bigm| \bigm|
\nu =0
, \chi 1(x) = - 1
\lambda \prime (\nu )
dR\nu (x)
d\nu
\bigm| \bigm| \bigm| \bigm|
\nu =1
. (3.7)
Note that here and in what follows the differentiation with respect to a natural parameter n \in \BbbN
means: 1) the switch to a real parameter \nu \in \BbbR , i.e., the use of the hypergeometrical or confluent
hypergeometrical functions, 2) the differentiation by \nu , 3) the substitution of n instead of \nu in the
derivative.
2. The set of functions
u0(x) = \chi 0(x) + c0P0(x) + d0Q0(x), u1(x) = \chi 1(x) + c1P1(x) + d1Q1(x) (3.8)
with arbitrary coefficients c0, c1, d0, d1 represents particular solutions of the inhomogeneous res-
onant equation too. These coefficients can be chosen at the next step of the algorithm so that the
following particular solutions uk(x), k = 2, 3, . . . , obtained by the recursion below satisfy the
corresponding resonant equation.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. I 197
3. Differentiating the recurrence equation (3.6) for Rn by n we obtain
un+1(x) = - 1
\lambda \prime (n+ 1)
\biggl[
- d\lambda (n)
dn
(\alpha nx+ \beta n)un(x) +
d\lambda (n - 1)
dn
\gamma nun - 1(x) +
+
\biggl(
d\alpha n
dn
x+
d\beta n
dn
\biggr)
Rn(x) -
d\gamma n
dn
Rn - 1(x)
\biggr]
, n = 1, 2, . . . . (3.9)
We set here n = 1 and demand that u2(x) obtained from (3.9), (3.8), satisfies the resonant differential
equation (3.5). From this condition we determine the coefficients c0, c1, d0, d1 and, therefore, the
initial values (3.8) for the recursive algorithm (3.9). Using Theorem 3.1 we prove below that un(x)
then satisfy the resonant equation for all n = 0, 1, 2, . . . .
4. Resonant equation of the Legendre type. 4.1. The Legendre resonant equation of the first
kind. Let us consider the following inhomogeneous equation with the Legendre differential operator
on the left-hand and the Legendre polynomial on the right-hand side:
d
dx
\biggl[
(1 - x2)
du(x)
dx
\biggr]
+ n(n+ 1)u(x) = Pn(x). (4.1)
It is the resonant equation of the first kind since the Legendre polynomial Pn(x) satisfies the corre-
sponding homogeneous differential equation. The second linear independent solution of the homo-
geneous differential equation Qn(x) is called the Legendre function of the second kind. The general
solution of the homogeneous differential equation (4.1) is given by
u(x) = c1Pn(x) + c2Qn(x),
where c1, c2 are arbitrary constants.
The explicit representation of the Legendre function of the second kind can be presented through
the hypergeometric function (see, e.g., [5], \S 10.10):
Qn(x) = Q0(x)Pn(x) -
\lfloor n+1
2
\rfloor \sum
k=1
2n - 4k + 3
(2k - 1)(n - k + 1)
Pn - 2k+1(x) =
=
2n(n!)2
(2n+ 1)!(1 + x)n+1
F
\biggl(
n+ 1, n+ 1; 2n+ 2;
2
1 + x
\biggr)
=
= ( - 1)n+1 2n(n!)2
(2n+ 1)!(1 - x)n+1
F
\biggl(
n+ 1, n+ 1; 2n+ 2;
2
1 - x
\biggr)
=
=
1
2
\biggl[
F
\biggl(
n+ 1, n+ 1; 2n+ 2;
2
1 + x
\biggr)
+
+( - 1)n+1 2n(n!)2
(2n+ 1)!(1 - x)n+1
F
\biggl(
n+ 1, n+ 1; 2n+ 2;
2
1 - x
\biggr) \biggr]
,
Q0(x) =
1
2
\mathrm{l}\mathrm{n}
x+ 1
x - 1
.
(4.2)
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198 I. GAVRILYUK, V. MAKAROV
Here F (a, b; c; z) =
\sum n
p=0
(a)p(b)pz
n
(c)pp!
is the hypergeometric function of z, (a)0 = 1, (a)p =
=
\Gamma (a+ p)
\Gamma (a)
is the Pochhammer symbol, and \Gamma (x) is the Gamma function. We remember, that many
of the well known mathematical functions can be expressed in terms of the hypergeometric function,
or as limiting cases of it. Two typical examples are
\mathrm{l}\mathrm{n}(1 + z) = z F (1, 1; 2; - z),
(1 - z) - a = F (a, 1; 1; z).
The Legendre functions as well as several orthogonal polynomials, including Jacobi polynomials
P
(\alpha ,\beta )
n and their special cases Legendre polynomials (\alpha = 0, \beta = 0), Chebyshev polynomials Tn(x)
(\alpha = - 1/2, \beta = - 1/2), Gegenbauer polynomials C\lambda
n(x) (\alpha = \beta = \lambda - 1/2) can be written in terms
of hypergeometric functions in many ways, for example (see, e.g., [7], \S 10.8),
P (\alpha ,\beta )
n (x) =
\biggl(
n+ \alpha
n
\biggr)
Fn
\biggl(
- n, n+ \alpha + \beta + 1;\alpha + 1;
1 - x
2
\biggr)
=
= ( - 1)n
\biggl(
n+ \beta
n
\biggr)
Fn
\biggl(
- n, n+ \alpha + \beta + 1;\beta + 1;
1 + x
2
\biggr)
,
Pn(x) = P (0,0)
n (x) =
1
2
\biggl[
Fn
\biggl(
- n, n+ 1; 1;
1 - x
2
\biggr)
+
+( - 1)n Fn
\biggl(
- n, n+ 1; 1;
1 + x
2
\biggr) \biggr]
.
(4.3)
The use of the hypergeometric functions to obtain a solution of a resonant equation represents
a direct way to solve the resonant equations. This way is due to the fact that the hypergeometric
differential equation
z(1 - z)
d2u
dz2
+ [c - (a+ b+ 1)z]
du
dz
- abu = 0 (4.4)
with an appropriate choice of their parameters can be transformed to the following Legendre equation
[4] (\S 3.2)
(1 - z2)
d2w
dz2
- 2z
dw
dz
+ \nu (\nu + 1)w = 0. (4.5)
Due to Theorem 3.1 and (4.3) we have a particular solution of (4.1) in the form
un(x) =
1
2
[\~un(x) + ( - 1)n\~un( - x)] , (4.6)
where (see (4.3) as well as [5], \S 10.8, formula (16) with \alpha = 0, \beta = 0)
\~un(x) = kn
d
d\nu
P\nu (x)| \nu =n = kn
d
d\nu
F\nu
\biggl(
1 + \nu , - \nu ; 1; 1 - x
2
\biggr) \bigm| \bigm| \bigm| \bigm|
\nu =n
=
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RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. I 199
= kn
\left[ n\sum
p=1
d
dn
(1 + n)p( - n)p
(p!)2
\biggl(
1 - x
2
\biggr) p
+
+( - 1)n+1n!
\infty \sum
p=n+1
(1 + n)p(p - n - 1)!
(p!)2
\biggl(
1 - x
2
\biggr) p
\right] (4.7)
with kn = - 1
2n+ 1
. The last formula can be transformed in the following way
\~un(x) = kn
\left[ n\sum
p=1
- 1
(p!)2
\Biggl[
2n+ p - 2n2(n+ p)
p - 1\sum
i=1
1
i2 - n2
\Biggr]
p - 1\prod
i=1
(i2 - n2)
\biggl(
1 - x
2
\biggr) p
+
+( - 1)n+1
\infty \sum
p=n+1
1
p
n\prod
i=1
p+ i
p - i
\biggl(
1 - x
2
\biggr) p
\right] . (4.8)
Using the formulas
1
p
n\prod
i=1
p+ i
p - i
=
n\sum
i=0
an,i
p - i
, an,i = ( - 1)n+i (n+ i)!
(n - i)!(i!)2
,
( - 1)n
n\sum
i=0
an,i
\biggl(
1 - x
2
\biggr) i
\equiv Fn
\biggl(
1 + n, - n; 1; 1 - x
2
\biggr)
= Pn(x) (4.9)
the sum of the last series can be transformed to
\infty \sum
p=n+1
1
p
n\prod
i=1
p+ i
p - i
\biggl(
1 - x
2
\biggr) p
=
= -
n\sum
i=0
an,i
\biggl(
1 - x
2
\biggr) i
\mathrm{l}\mathrm{n}
\biggl(
1 + x
2
\biggr)
-
n - 1\sum
p=0
an,p
\biggl(
1 - x
2
\biggr) p n - p\sum
i=1
1
i
\biggl(
1 - x
2
\biggr) i
=
= ( - 1)n+1Pn(x) \mathrm{l}\mathrm{n}
\biggl(
1 + x
2
\biggr)
-
n\sum
i=1
\biggl(
1 - x
2
\biggr) i i - 1\sum
p=0
an,p
i - p
. (4.10)
Thus, for the function (4.7) we have
\~un(x) = - 1
2n+ 1
Pn(x) \mathrm{l}\mathrm{n}
\biggl(
1 + x
2
\biggr)
+
n\sum
i=1
\biggl(
1 - x
2
\biggr) i
bn,i,
bn,i = - 1
2n+ 1
\left[ 1
i!
d
dn
(1 + n)i( - n)i + ( - 1)n
i - 1\sum
p=0
an,p
i - p
\right] .
(4.11)
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200 I. GAVRILYUK, V. MAKAROV
Such direct way to obtain a particular solution as described above is rather awkward. Below we
propose an algorithmic way based on Theorem 3.1 and on the recursion formula for the corresponding
orthogonal polynomials. This algorithm can be easily implemented by the computer algebra tools,
for example, by Maple.
Actually, for n = 0, 1 using Theorem 3.1 we obtain from (4.6) the following particular solutions:
\chi 0(x) = - 1
2
\mathrm{l}\mathrm{n}(1 - x2), \chi 1(x) = - x
6
\mathrm{l}\mathrm{n}(1 - x2) +
11
18
x. (4.12)
Differentiating the recurrence relation for Pn(x) by n we arrive at the following recurrence equation
for particular solutions:
un+1(x) = - 1
2n+ 3
\biggl[
- (2n+ 1)2x
n+ 1
un(x) +
n(2n - 1)
n+ 1
un - 1(x) +
+
x
(n+ 1)2
Pn(x) -
1
(n+ 1)2
Pn - 1(x)
\biggr]
, n = 1, 2, . . . . (4.13)
The Legendre polynomials P0(x) = 1 and P1(x) = x as well as the Legendre functions of the second
kind Q0(x) and Q1(x) satisfy the corresponding homogeneous Legendre differential equation, that’s
why in accordance with our algorithm and in regard of (4.12) we can use the ansatzes for the initial
values
u0(x) = - 1
2
\mathrm{l}\mathrm{n}(1 - x2) + c0P0(x) + d0Q0(x),
u1(x) = - x
6
\mathrm{l}\mathrm{n}(1 - x2) +
11
18
x+ c1P1(x) + d1Q1(x)
(4.14)
with undefined coefficients c0, c1, d0, d1. After substituting these into (4.13) with n = 1 we demand
that u2(x) satisfies the resonant differential equation (4.1), from where we obtain
d0 = 0, d1 = 0,
c0 =
17
6
+ 3c1.
(4.15)
Setting, for example, c0 = 0 we obtain c1 = - 17
18
and arrive at the representations
u1(x) = - 1
6
P1(x) \mathrm{l}\mathrm{n} (1 - x2) - 1
3
x,
u2(x) = - 1
10
P2(x) \mathrm{l}\mathrm{n} (1 - x2) - 7
20
x2 +
1
20
.
(4.16)
In general, we have
un(x) = - 1
2(2n+ 1)
Pn(x) \mathrm{l}\mathrm{n} (1 - x2) + vn(x), (4.17)
where vn(x) satisfies the recurrence equation
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RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. I 201
vn+1(x) = - 1
2n+ 3
\biggl[
- (2n+ 1)2x
n+ 1
vn(x) +
n(2n - 1)
n+ 1
vn - 1(x) +
+
x
(n+ 1)2
Pn(x) -
1
(n+ 1)2
Pn - 1(x)
\biggr]
, n = 1, 2, . . . ,
v0(x) = 0, v1(x) = - x
3
.
(4.18)
This recurrence equation together with (4.14), (4.17) provides, for example, the following particular
solutions:
u3(x) = - 1
14
P3(x) \mathrm{l}\mathrm{n} (1 - x2) +
5
28
x - 37
84
x3,
u4(x) = - 1
18
P4(x) \mathrm{l}\mathrm{n} (1 - x2) - 7
288
+
59
144
x2 - 533
864
x4.
(4.19)
The next theorem shows that the functions un(x) obtained by our recursive algorithm satisfy the
resonant Legendre differential equation of the first kind for all n = 0, 1, . . . .
Theorem 4.1. The functions un(x) obtained by the recursive algorithm (4.13) satisfy the reso-
nant Legendre differential equation of the first kind (4.1) for each n = 0, 1, 2, . . . .
Proof. These functions for n = 0, 1, 2 satisfy the resonant Legendre differential equation by
construction. Let us assume that up(x), p = 0, 1, . . . , n, satisfy this differential equation and prove
that it is the case for p = n+ 1. Differentiating the classical relation [7] (\S 10.10)
(1 - x2)
dPn(x)
dx
= n
\bigl[
Pn - 1(x) - xPn(x)
\bigr]
(4.20)
by n and using Theorem 3.1 we arrive at the equation
- (2n+ 1)(1 - x2)
dun(x)
dx
= - n (2n - 1)un - 1(x) + xn (2n+ 1)un(x) + Pn - 1(x) - xPn(x).
(4.21)
Applying to (4.13) the Legendre differential operator
\scrA n = (1 - x2)
d2
dx2
- 2x
d
dx
+ n(n+ 1) (4.22)
we obtain
\scrA n+1un+1(x) = Pn+1(x) +
2(2n+ 1)
(2n+ 3)(n+ 1)
\biggl[
(2n+ 1)(1 - x2)
dun(x)
dx
-
- n (2n - 1)un - 1(x) + xn (2n+ 1)un(x) + Pn - 1(x) - xPn(x)
\biggr]
. (4.23)
It follows from (4.21) that the expression in the square brackets is equal to zero which proves the
theorem.
Now, the general solution of the inhomogeneous equation (4.1) can be represented by
u(x) = uh + un(x) = c1Pn(x) + c2Qn(x) + un(x), (4.24)
where c1, c2 are arbitrary constants.
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202 I. GAVRILYUK, V. MAKAROV
Remark 4.1. The basis of the proof of Theorem 4.1 are a recurrence equation xpn(x) =
= \alpha npn - 1(x) + \beta npn(x) + \gamma npn - 1(x) for the corresponding orthogonal with the weight \sigma (x) poly-
nomial pn(x) and the differentiation formula \sigma (x)p\prime n(x) = \alpha
(1)
n pn+1(x) + (\beta
(1)
n + \gamma
(1)
n x)pn(x) (see
(4.20) for the Legendre polynomials) which represents the weighted derivative of the polynomial
under consideration through two neighboring polynomials (see, e.g., [22], \S 9). But the second linear
independent solution of the corresponding homogeneous equation, which is called the function of the
second kind (in the case above, the Legendre function of the second kind Qn(x), which is not poly-
nomial!) satisfies the same recurrence equation and the same differentiation formula (see [22, p. 67]).
Thus, a similar theorem for the particular solutions of the resonant equations of the second kind
obtained by the corresponding recursive algorithm is valid too.
4.2. The Legendre resonant equation of the second kind. In this subsection we consider the
equation (4.1) with the Legendre function of the second kind
Qn(x) =
2n(1 + x) - n - 1(n!)2
(2n+ 1)!
F
\biggl(
n+ 1, n+ 1; 2n+ 2;
2
1 + x
\biggr)
as the right-hand side, i.e., we have again a resonant equation.
The general solution of such resonant equation is
u(x) = c1Pn(x) + c2Qn(x) + un(x), (4.25)
where the linear independent Legendre polynomial Pn(x) and the Legendre function of the second
kind Qn(x) satisfy the homogeneous Legendre equation, c1, c2 are arbitrary constants and un(x) is
a particular solution of the inhomogeneous resonant equation.
Note, that in [7] a solution is obtained for the case n = 0 only and there was pointed out that it
is very difficult to obtain the solution for other n in a closed form. But our Theorem 3.1 allows one
to obtain the particular solution for arbitrary n as
un(x) = - 1
(2n+ 1)
[2\psi (n+ 1) - 2\psi (2n+ 2) + \mathrm{l}\mathrm{n}(2) - \mathrm{l}\mathrm{n} (1 + x)]Qn(x) -
- 2n(n!)2(1 + x) - n - 1
(2n+ 1)(2n+ 1)!
d
d\nu
\biggl[
F
\biggl(
\nu + 1, \nu + 1; 2\nu + 2;
2
1 + x
\biggr) \biggr]
\nu =n
, (4.26)
where \psi (z) =
d
dz
\mathrm{l}\mathrm{n} \Gamma (z) =
\Gamma \prime (z)
\Gamma (z)
is the logarithmic derivative of the Gamma function. Various
representations of this function can be found in [5] (\S 1.7). For n = 0, 1 we obtain the particular
solutions
\chi 0(x) = - P0(x)w(x),
\chi 1(x) = - 1
3
P1(x)w(x) - 1
6
\mathrm{l}\mathrm{n}
\bigl(
x2 - 1
\bigr)
- 2
3
,
w(x) = - \mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
2,
2
1 + x
\biggr)
- 1
2
\mathrm{l}\mathrm{n}2 (x+ 1) +
1
2
\mathrm{l}\mathrm{n} (x+ 1) \mathrm{l}\mathrm{n} (x - 1) =
= - \mathrm{d}\mathrm{i}\mathrm{l}\mathrm{o}\mathrm{g}
\biggl(
2
1 + x
\biggr)
- 1
2
\mathrm{l}\mathrm{n}2 (x+ 1) +
1
2
\mathrm{l}\mathrm{n} (x+ 1) \mathrm{l}\mathrm{n} (x - 1) , x > 1, (4.27)
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RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. I 203
where polylog is the so-called polylogarithm-function of order s and of the argument z (Jonquiere’s
function):
polylog(s, z) = Lis(z) =
\infty \sum
k=1
zk
ks
(dilog or Spence’s function, denoted also by Li2(z) is a special case of polylog for s = 2).
Other explicit representations of particular solutions can be obtain by Algorithm 3.1. Differenti-
ating the recurrence relation for the Legendre function of the second kind
Q\nu +1(x) =
x(2\nu + 1)
(\nu + 1)
Q\nu (x) -
\nu
(\nu + 1)
Q\nu - 1(x) (4.28)
with respect to \nu and taking into account Theorem 3.1 we obtain the recurrence formula
un+1(x) = - 1
2n+ 3
\biggl[
- (2n+ 1)2x
n+ 1
un(x) +
n(2n - 1)
n+ 1
un - 1(x) +
x
(n+ 1)2
Qn(x) -
- 1
(n+ 1)2
Qn - 1(x)
\biggr]
, n = 1, 2, . . . . (4.29)
In accordance with our algorithm and in regard of (4.27) we use the following ansatzes for initial
values:
u0(x) = - P0(x)w(x) + c0P0(x) + d0Q0(x),
u1(x) = - 1
3
P1(x)w(x) -
1
6
\mathrm{l}\mathrm{n}
\bigl(
x2 - 1
\bigr)
- 2
3
+ c1P1(x) + d1Q1(x), x > 1,
(4.30)
with undefined coefficients c0, c1, d0, d1. After substitution into (4.29) with n = 1 we demand that
u2(x) satisfies the resonant differential equation of the second kind. Then we obtain
c0 = 0, c1 = 0,
- d0 + 3d1 + 1 = 0.
(4.31)
Setting, for example, d0 = - 1
2
we get d1 = - 1
2
and herewith the particular solution
u2(x) = - 1
5
P2(x)w(x) -
3x
20
\mathrm{l}\mathrm{n}(x2 - 1) - 1
30
\mathrm{l}\mathrm{n}
\biggl(
x+ 1
x - 1
\biggr)
- 3x
5
- 1
3
Q2(x), x > 1. (4.32)
The particular solutions un(x), n = 3, 4, . . . , can be obtained using (4.29) and the initial conditions
(4.30), (4.32).
The next theorem shows that the functions un(x) obtained by our recursive algorithm satisfy the
resonant Legendre differential equation of the second kind for all n = 0, 1, . . . .
Theorem 4.2. The functions un(x) obtained by the recursive algorithm (4.29) satisfy the reso-
nant Legendre differential equation of the second kind (4.1) for each n = 0, 1, 2, . . . .
The proof is completely analogous to that of Theorem 4.1 in regard of the fact that the Legendre
functions of the second kind (which are not polynomials!) satisfy the same recurrence equation as
the Legendre polynomials and the differentiating formula (4.20) (see [7], \S 10.10).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
204 I. GAVRILYUK, V. MAKAROV
5. Resonant equation of the Jacobi type. 5.1. The Jacobi resonant equation of the first kind.
In this section we consider the resonant equation of the Jacobi type
(1 - x2)
d2un(x)
dx2
+
+[\beta - \alpha - (\alpha + \beta + 2)x]
dun(x)
dx
+ n(n+ \beta + 1)un(x) = P (\alpha ,\beta )
n (x), (5.1)
where P (\alpha ,\beta )
n (x) is the Jacobi polynomial [5] (\S 10.8) satisfying the homogeneous differential equa-
tion. The general solution of this equation is
u(x) = c1P
(\alpha ,\beta )
n (x) + c2Q
(\alpha ,\beta )
n (x) + un(x), (5.2)
where
Q(\alpha ,\beta )
n (x) =
2n+\alpha +\beta \Gamma (n+ \alpha + 1)\Gamma (n+ \beta + 1)
(x - 1)n+\alpha +1(x+ 1)\beta \Gamma (2n+ \alpha + \beta + 2)
\times
\times F
\biggl(
n+ 1, n+ \alpha + 1; 2n+ \alpha + \beta + 2;
2
1 - x
\biggr)
(5.3)
is the Jacobi function of the second kind [5] (\S 10.8), c1, c2 are arbitrary constants and un(x) is a
particular solution of the inhomogeneous equation.
Due to Theorem 3.1 we have for a particular solution
un(x) = - 1
2n+ \alpha + \beta + 1
\biggl[
\partial
\partial \nu
P (\alpha ,\beta )
\nu (x)
\biggr]
\nu =n
=
= - 1
2n+ \alpha + \beta + 1
\biggl[
\partial
\partial \nu
\Gamma (\nu + \alpha )
\Gamma (\alpha )\Gamma (\nu + 1)
F
\biggl(
- \nu , \nu + \alpha + \beta + 1;\alpha + 1;
1 - x
2
\biggr) \biggr]
\nu =n
=
= - 1
2n+ \alpha + \beta + 1
\Biggl\{
[\Psi (n+ \alpha ) - \Psi (n+ 1)]P (\alpha ,\beta )
n (x)+
+
\Gamma (n+ \alpha )
\Gamma (\alpha )\Gamma (n+ 1)
\partial
\partial \nu
F
\biggl(
- \nu , \nu + \alpha + \beta + 1;\alpha + 1;
1 - x
2
\biggr) \bigm| \bigm| \bigm| \bigm|
\nu =n
\Biggr\}
=
= - 1
2n+ \alpha + \beta + 1
\left\{ [\Psi (n+ \alpha ) - \Psi (n+ 1)]P (\alpha ,\beta )
n (x)+
+
\Gamma (n+ \alpha )
\Gamma (\alpha )\Gamma (n+ 1)
\left[ n\sum
p=1
d
dn
(\alpha + \beta + 1 + n)p( - n)p
p!(\alpha + 1)p
\biggl(
1 - x
2
\biggr) p
+
+( - 1)n+1n!
\infty \sum
p=n+1
(\alpha + \beta + 1 + n)p(p - n - 1)!
p!(\alpha + 1)p
\biggl(
1 - x
2
\biggr) p
\right] \right\} . (5.4)
For example, we have the particular solutions
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RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. I 205
\chi 0(x) = u0(x) =
1
\alpha + \beta + 1
\left[ - \psi (\alpha ) + \psi (1) +
\infty \sum
p=1
(\alpha + \beta + 1)p
p(\alpha + 1)p
\biggl(
1 - x
2
\biggr) p
\right] ,
\chi 1(x) = u1(x) = - 1
\alpha + \beta + 3
\Biggl[
(\psi (\alpha + 1) - \psi (2))P
(\alpha ,\beta )
1 (x) -
(5.5)
- \alpha (\alpha + \beta + 3)
(\alpha + 1)
\biggl(
1 - x
2
\biggr)
+ \alpha
\infty \sum
p=2
(\alpha + \beta + 2)p
p(p - 1)(\alpha + 1)p
\biggl(
1 - x
2
\biggr) p
\Biggr]
.
Differentiating (with respect to n) the recurrence formula for the Jacobi polynomials
P
(\alpha ,\beta )
n+1 (x) = (a(n)x+ b(n))P (\alpha ,\beta )
n (x) - c(n)P
(\alpha ,\beta )
n - 1 (x),
a(n) =
(2n+ \alpha + \beta + 1)(2n+ \alpha + \beta + 2)
2(n+ 1)(n+ \alpha + \beta + 1)
,
b(n) =
\alpha 2 - \beta 2
2(n+ 1)(n+ \alpha + \beta + 1)(2n+ \alpha + \beta )
,
c(n) =
(n+ \alpha )(n+ \beta )(2n+ \alpha + \beta + 2)
(n+ 1)(n+ \alpha + \beta + 1)(2n+ \alpha + \beta )
(5.6)
with taking into account (5.4) we arrive at the recursion
un+1(x) = - 1
2n+ \alpha + \beta + 3
\Bigl[
- (2n+ \alpha + \beta + 1)(a(n)x+ b(n))un(x)+
+(2n+ \alpha + \beta - 1)c(n)un - 1(x) + (a\prime (n)x+ b\prime (n))P (\alpha ,\beta )
n (x) - c\prime (n)P
(\alpha ,\beta )
n - 1 (x)
\Bigr]
, (5.7)
n = 1, 2, . . . ,
and with the initial conditions (5.5) we can obtain un(x) for arbitrary n.
It is rather complicated to obtain an explicit formula for the solution of the Jacobi resonant
equation for arbitrary \alpha , \beta , therefore we consider an example only.
Example 5.1. Let us consider the case of the Jacobi resonant equation of the first kind with
\alpha = 1, \beta = 2. From (5.4) with n = 0, 1 we have the particular solutions
\chi 0(x) = - 1
64
[5 \mathrm{l}\mathrm{n} (x+ 1) + 11 \mathrm{l}\mathrm{n} (x - 1)]+
+
5(x - 1) + 10(x2 - 1) - 11 (x+ 1)2
96 (x+ 1)2 (x - 1)
,
\chi 1(x) = - 5x - 1
384
[10 \mathrm{l}\mathrm{n} (x+ 1) + 17 \mathrm{l}\mathrm{n} (x - 1)]+
+
1075x4 + 1298x3 - 1842x2 - 1918x+ 487
96 (x+ 1)2 (x - 1)
.
(5.8)
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206 I. GAVRILYUK, V. MAKAROV
The initial values for recursion (5.7) are chosen in the form
u0(x) = \chi 0(x) + d0Q
(1,2)
0 (x) + c0P
(1,2)
0 (x),
u1(x) = \chi 1(x) + d1Q
(1,2)
1 (x) + c1P
(1,2)
1 (x), x > 1,
(5.9)
where the undefined coefficients are determined so that u2(x) satisfies the resonant differential equa-
tion. We substitute (5.9) into (5.1) with n = 1 and then into the resonant differential equation, which
yields
c0 = 0, d0 = - 7
24
,
c1 = - 47
2880
, d1 = 0
(5.10)
and further proceed in accordance with our Algorithm 3.1.
5.2. The Jacobi resonant equation of the second kind. In this subsection we consider the
resonant equation
(1 - x2)
d2u(x)
dx2
+ [\beta - \alpha - (\alpha + \beta + 2)x]
du(x)
dx
+
+n(n+ \alpha + \beta + 1)u(x) = Q(\alpha ,\beta )
n (x), (5.11)
where Q(\alpha ,\beta )
n (x) is the Jacobi function of the second kind [5] (\S 10.8) given by formula (5.3).
Due to Theorem 3.1 we have for a particular solution the general formula
un(x) = - 1
2n+ \alpha + \beta + 1
\biggl[
\partial
\partial \nu
Q(\alpha ,\beta )
\nu (x)
\biggr]
\nu =n
=
= - 1
2n+ \alpha + \beta + 1
\biggl\{
[\mathrm{l}\mathrm{n}(2) + \Psi (n+ \alpha + 1) + \Psi (n+ \beta + 1) - \mathrm{l}\mathrm{n}(x - 1) -
- 2\Psi (2n+ \alpha + \beta + 1)]Q(\alpha ,\beta )
n (x)+
+
2n+\alpha +\beta \Gamma (n+ \alpha + 1)\Gamma (n+ \beta + 1)
(x - 1)n+\alpha +1(x+ 1)\beta \Gamma (2n+ \alpha + \beta + 2)
\times
\times \partial
\partial \nu
F
\biggl(
\nu + 1, \nu + \alpha + 1; 2\nu + \alpha + \beta + 1;
2
1 - x
\biggr) \bigm| \bigm| \bigm| \bigm|
\nu =n
\biggr\}
. (5.12)
This formula is rather complicated for practical use, therefore we use our recursive algorithm.
By differentiation of the recurrence equation we obtain the following recurrence formula:
un+1(x) = - 1
2n+ \alpha + \beta + 3
\Bigl[
- (2n+ \alpha + \beta + 1)(a(n)x+ b(n))un(x)+
+(2n+ \alpha + \beta - 1)c(n)un - 1(x) + (a\prime (n)x+ b\prime (n))Q(\alpha ,\beta )
n (x) - c\prime (n)Q
(\alpha ,\beta )
n - 1 (x)
\Bigr]
, (5.13)
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RESONANT EQUATIONS WITH CLASSICAL ORTHOGONAL POLYNOMIALS. I 207
n = 1, 2, . . . ,
which together with \chi 0(x) = u0(x), \chi 1(x) = u1(x) from (5.12) and with the corresponding ansatz
for the initial values provides an algorithm for un(x) for any n = 2, 3, . . . . Since the formulas in the
general case are rather cumbersome, we restrict ourself to an example.
Example 5.2. Let \alpha = 1, \beta = 2, then we have for the Jacobi functions of the second kind
Q
(1,2)
0 (x) = - 1
2
\mathrm{l}\mathrm{n}
\biggl(
x+ 1
x - 1
\biggr)
+
3x2 + 3x - 2
3(x+ 1)2(x - 1)
,
Q
(1,2)
1 (x) = - 5x - 1
4
\mathrm{l}\mathrm{n}
\biggl(
x+ 1
x - 1
\biggr)
+
15x3 + 12x2 - 13x - 8
6(x+ 1)2(x - 1)
,
Q
(1,2)
2 (x) = - 21x2 - 6x - 3
8
\mathrm{l}\mathrm{n}
\biggl(
x+ 1
x - 1
\biggr)
+
105x4 + 75x3 - 115x2 - 65x+ 16
20(x+ 1)2(x - 1)
, . . . .
(5.14)
The general formula is the following:
Q(1,2)
n (x) = - P (1,2)
n (x)
1
2
\mathrm{l}\mathrm{n}
\biggl(
x+ 1
x - 1
\biggr)
+ q(1,2)n (x), (5.15)
where the functions q(1,2)n (x) satisfy the recurrence equation for the Jacobi polynomials but with the
initial conditions, which are given by the second summands in Q(1,2)
0 (x), Q
(1,2)
1 (x). From (5.12) we
obtain the following particular solutions of the resonant equation of the second kind for n = 0, 1
\chi 0(x) = - 1
4
w(x) +
15x - 11
48(x - 1)
\mathrm{l}\mathrm{n}(x+ 1) - 15x2 + 22x+ 3
48(x+ 1)2
\mathrm{l}\mathrm{n}(x - 1) -
- 9x2 + 3x - 4
72(x+ 1)2 (x - 1)
,
\chi 1(x) = - 1
12
(5x - 1)w(x)+
+
- 135x3 - 159x2 + 42x+ 56
360(x+ 1)2
\mathrm{l}\mathrm{n}(x - 1) +
45x2 - 32x - 8
120(x - 1)
\mathrm{l}\mathrm{n}(x+ 1)+
+
22025x4 + 18340x3 - 25422x2 - 18452x+ 3413
8640 (x+ 1)2 (x - 1)
, (5.16)
w(x) = dilog
\biggl(
x+ 1
2
\biggr)
+
1
2
\mathrm{l}\mathrm{n} (x+ 1) \mathrm{l}\mathrm{n}(x - 1) - \mathrm{l}\mathrm{n}(2) \mathrm{l}\mathrm{n}(x - 1).
For the initial values in our recursive algorithm we use the ansatzes
u0(x) = \chi 0(x) + c0P
(1,2)
0 (x) + d0Q
(1,2)
0 (x),
u1(x) = \chi 1(x) + c1P
(1,2)
1 (x) + d1Q
(1,2)
1 (x), x > 1,
(5.17)
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 2
208 I. GAVRILYUK, V. MAKAROV
with undefined coefficients c0, d0, c1, d1. Substituting these into (5.13) with n = 1 and demanding
that the result satisfies the resonant differential equation, we obtain
d0 =
3
2
c0 +
7
80
, d1 =
3
2
c1 +
881
516
(5.18)
and herewith the particular solution of the resonant equation
u(1,2)n (x) = - 1
2n+ 4
P (1,2)
n (x)w(x) + p(1,2)n (x) \mathrm{l}\mathrm{n}(x+ 1) + r(1,2)n (x) \mathrm{l}\mathrm{n}(x - 1) + v(1,2)n (x). (5.19)
Here the functions p(1,2)n (x), r
(1,2)
n (x) satisfy the recurrence relation for the Jacobi polynomials with
the initial conditions
p
(1,2)
0 (x) =
15x - 11
48(x - 1)
, p
(1,2)
1 (x) =
45x2 - 32x - 8
120(x - 1)
,
r
(1,2)
0 (x) = - 15x2 + 22x+ 3
48(x+ 1)2
, r
(1,2)
1 (x) =
- 135x3 - 159x2 + 42x+ 56
360(x+ 1)2
.
(5.20)
The function v(1,2)n (x) satisfies the recurrence equation
vn+1(x) = - 1
2n+ \alpha + \beta + 3
\Bigl[
- (2n+ \alpha + \beta + 1)(a(n)x+
+b(n))vn(x) + (2n+ \alpha + \beta - 1)c(n)vn - 1(x) +
+(a\prime (n)x+ b\prime (n))Q(\alpha ,\beta )
n (x) - c\prime (n)Q
(\alpha ,\beta )
n - 1 (x)
\Bigr]
, (5.21)
n = 1, 2, . . . ,
with \alpha = 1, \beta = 2 and with the initial conditions
v0(x) = - 9x2 + 3x - 4
72(x+ 1)2 (x - 1)
+ d0Q
(1,2)
0 (x),
v1(x) =
22025x4 + 18340x3 - 25422x2 - 18452x+ 3413
8640 (x+ 1)2 (x - 1)
+ d1Q
(1,2)
1 (x).
(5.22)
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Received 21.11.18
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| id | umjimathkievua-article-1431 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:13Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/be/0cef773a87c21da098f007eb0fe5c9be.pdf |
| spelling | umjimathkievua-article-14312019-12-05T08:54:43Z Resonant equations with classical orthogonal polynomials. I Резонанснi рiвняння з класичними ортогональними полiномами. I Gavrilyuk, I. P. Makarov, V. L. Гаврилюк, І. П. Макаров, В. Л. In the present paper, we study some resonant equations related to the classical orthogonal polynomials and propose an algorithm of finding their particular and general solutions in the explicit 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 aimed at 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 for the square operator equations $A^2u = f$; e.g., for the biharmonic equation. Вивчаються деякi резонанснi рiвняння, що мають вiдношення до класичних ортогональних полiномiв. Запропо- новано алгоритм знаходження їхнiх частинних та загальних розв’язкiв у явному виглядi. Цей алгоритм найкрaще пiдходить для методiв комп’ютерної алгебри, таких як Maple. Резонанснi рiвняння складають суттєву частину багатьох застосувань, зокрема ефективного функцiонально-дискретного методу, що застосовується при розв’язаннi операторних рiвнянь та задач на власнi значення. Такi рiвняння також з’являються в контекстi суперсиметричних операторiв Казимiра для дiспiнової алгебри, а також для квадратичних операторних рiвнянь $A^2u = f$, наприклад для бiгармонiчного рiвняння. Institute of Mathematics, NAS of Ukraine 2019-02-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1431 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 2 (2019); 190-209 Український математичний журнал; Том 71 № 2 (2019); 190-209 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1431/415 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. I |
| title | Resonant equations with classical orthogonal polynomials. I |
| title_alt | Резонанснi рiвняння з класичними
ортогональними полiномами. I |
| title_full | Resonant equations with classical orthogonal polynomials. I |
| title_fullStr | Resonant equations with classical orthogonal polynomials. I |
| title_full_unstemmed | Resonant equations with classical orthogonal polynomials. I |
| title_short | Resonant equations with classical orthogonal polynomials. I |
| title_sort | resonant equations with classical orthogonal polynomials. i |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1431 |
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