Received: 23 July 2017 DOI: 10.1002/mma.4751
RESEARCH ARTICLE
Nonlinear difference equations arising from the generalized Stieltjes-Wigert and q-Laguerre weights Hongmei Chen1
Galina Filipuk2
1 Faculty of Science and Technology, Department of Mathematics, University of Macau, Av. Padre Tomás Pereiro Taipa, Macau, China 2
Faculty of Mathematics, Informatics and Mechanics, University of Warsaw, Banacha 2 Warsaw, 02-097, Poland Correspondence Hongmei Chen, Faculty of Science and Technology, Department of Mathematics, University of Macau, Av. Padre Tomás Pereiro, Taipa, Macau, China. Email:
[email protected] Communicated by: V. Didenko
Yang Chen1
In this paper, we investigate the generalized Stieltjes-Wigert and q-Laguerre polynomials. We derive the second- and the third-order nonlinear difference equations for the subleading coefficients of these polynomials and use them to find a few terms of the formal expansions in powers of qn/2 . We also show how the recurrence coefficients in the three-term recurrence relation for these polynomials can be computed efficiently by using the nonlinear difference equations for the subleading coefficient. Moreover, we obtain systems of difference equations with one of the equations being q-discrete Painlevé III or V equations and analyze them by a singularity confinement. We also discuss certain generalized weights. K E Y WO R D S asymptotic expansions, difference equations, discrete Painlevé equations, orthogonal polynomials,
Funding information National Science Center (Poland), Grant/Award Number: OPUS 2017/25/B/BST1/00931 ; Science and Technology Development Fund of the Macau SAR, Grant/Award Number: FDCT 077/2012/A3, FDCT 130/2014/A3 and FDCT 023/2017/A1 ; University of Macau, Grant/Award Number: MYRG 201400011 FST and MYRG 201400004 FST
singularity confinement
MSC Classification: 33C47; 39A99; 34M55
1
I N T RO DU CT ION
Orthogonal polynomials appear in many areas of mathematics and mathematical physics, such as approximation theory, stochastic processes, and random matrix theory (see, for instance, previous studies1-3 ). Let {Pn (x)}, n ∈ N, be a sequence of monic polynomials of degree n in x, where Pn (x) = xn + 𝛿n xn−1 + ... + Pn (0),
(1)
which are orthogonal with respect to a positive weight w(x) on R+ ∞
∫0
Pm (x)Pn (x)w(x)dx =
𝛿m,n 𝛾n2
,
(2)
where 𝛿 m,n is the Kronecker delta. One of the most important properties of orthogonal polynomials is a three-term recurrence relation Math Meth Appl Sci. 2018;1–24.
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1
2
CHEN ET AL.
xPn (x) = Pn+1 (x) + bn Pn (x) + a2n Pn−1 (x),
n ≥ 0,
(3)
where P−1 (x) = 0 and P0 (x) = 1 and the coefficients a2n and bn are given by the integrals ∞
bn = 𝛾n2
∫0
∞
xPn (x)2 w(x)dx,
2 a2n = 𝛾n−1
∫0
xPn−1 (x)Pn (x)w(x)dx.
(4)
From the three-term recurrence relation (3), we have bn = 𝛿n − 𝛿n+1
(5)
and 𝛿 0 = 0, 𝛿 1 = −b0 . We also remark that the recurrence coefficients in the three-term recurrence relation can be expressed in terms of determinants whose entries are given in terms of the moments of the weight function.1 Orthogonal polynomials whose recurrence coefficients are related to solutions of either discrete or differential Painlevé equations are of particular importance (see, for instance, other studies4-7 and numerous references therein). It is known8 that the recurrence coefficients of the generalized Stieltjes-Wigert and q-Laguerre polynomials can be expressed in terms of the solutions of discrete Painlevé III and V equations, respectively. In particular, in Boelen and Van Assche,8 , Th. 1.1 the authors considered the generalized Stieltjes-Wigert weight w(x) =
(−x2 ; q2 )
x𝛼 , x ∈ [0, ∞), 𝛼 ∈ R, 2 2 2 ∞ (−q ∕x ; q )∞
(6)
and proved that the recurrence coefficients were related to the solutions xn of the q-discrete Painlevé III equation xn−1 xn+1 =
(xn + q−𝛼 )2 (qn+𝛼 xn + 1)2
(7)
by the following formulas: q2n+𝛼 b2n xn = xn+1 + q2n+2𝛼 xn−1 (xn + q−n−𝛼 )2 + 2(xn + q−𝛼 ), n ≥ 1,
(8)
and a2n = q1−n xn + q−2n−𝛼+1 , n ≥ 0. Moreover, x0 = −q−𝛼 , x1 = −b20 , and bn is to be taken positive. Similarly, the result in Boelen and Van Assche8 , Th. 1.2 relates the recurrence coefficients of orthogonal polynomials for the generalized q-Laguerre weight w(x) =
x𝛼 (−𝑝∕x2 ; q2 )∞ , x ∈ [0, ∞), 𝑝 ∈ [0, q−𝛼 ), 𝛼 ≥ 0, (−x2 ; q2 )∞ (−q2 ∕x2 ; q2 )∞
with the solutions zn of the q-discrete Painlevé V equation (zn zn−1 − 1)(zn zn+1 − 1) =
(zn +
√ √ q2−𝛼 ∕𝑝)2 (zn + 𝑝q𝛼−2 )2 . √ (qn+𝛼∕2−1 𝑝zn + 1)2
(9)
(10)
The explicit formulas8 for the recurrence coefficients are √ b2n q2n+𝛼 zn2 = zn zn+1 − 1 + q2n+2𝛼 ( 𝑝q−2−𝛼 zn + q−n−𝛼 )2 (zn zn−1 − 1) √ √ + 2(zn + q2−𝛼 ∕𝑝)(zn + 𝑝q𝛼−2 ) and
(11)
√ a2n = q−n+1 zn 𝑝q−2−𝛼 + q−2n−𝛼+1 . √ The initial conditions are given by z0 = − q2−𝛼 ∕𝑝 and z1 is expressed in terms of the first 2 moments of the weight w. We also remark that z1 can be expressed in terms of the coefficient b0 as follows:
CHEN ET AL.
3
b20 + 𝑝q−2 z1 = − √ . 𝑝q−2−𝛼 The results in Boelen and Van Assche8 were obtained by using the technique of ladder operators by Chen and Ismail.9 It allows one to use certain compatibility conditions for auxiliary functions An (x) and Bn (x) expressed as rational functions with unknown coefficients that depend on n. A skillful elimination of these coefficients (which is usually done case by case) leads to a second-order difference equation that is then identified with one of the discrete Painlevé equations. Recently, there have been numerous studies to derive nonlinear difference equations for recurrence coefficients and auxiliary quantities that appear in the definition of ladder operators for various weights (see, eg, literature9-13 and the references therein). The subleading coefficient is particularly useful in deriving Painlevé equations (eg, Chen and Pruessner14 ) for the recurrence coefficients. The main result of this paper is the derivation of nonlinear difference equations for the subleading coefficient 𝛿 n (see (1) of monic orthogonal polynomials for the weights considered in Boelen and Van Assche.8 We use these difference equations to find a few terms in the formal asymptotic expansion of the subleading coefficient. The finding of a true asymptotics is a subject for further studies. We note that there have been recent results on asymptotics of nonlinear q-difference Painlevé equations.15-17 We also use a difference equation for the subleading coefficient to compute the recurrence coefficient bn efficiently. Moreover, nonlinear difference equations derived in this paper give examples of discrete integrable equations. There are many papers on integrability of difference equations of second- and higher-order, methods to detect integrability in equations and various applications (see, for instance, Grammaticos et al18 ). This paper is organized as follows. In Section 2, we review the definition and basic properties of ladder operators for monic orthogonal polynomials on the real line and give supplementary conditions that will be used in the derivation of nonlinear difference equations. We illustrate the theory of ladder operators by using the classical Stieltjes-Wigert and q-Laguerre polynomials. In Section 3, we study the generalized Stieltjes-Wigert polynomials in detail. We show that the recurrence coefficients can be computed explicitly in terms of the subleading coefficient 𝛿 n . We derive nonlinear difference equations of orders 2 and 3 and a system of nonlinear difference equations involving 𝛿 n and solutions of discrete Painlevé equations. By using nonlinear difference equations, we obtain a few terms of a formal asymptotic expansion of the subleading coefficient. We also study a singularity confinement of the system of difference equations. In addition, we show how to compute the recurrence coefficients bn by using one of the nonlinear difference equations. In Section 4, we present similar results for polynomials orthogonal with respect to the generalized q-Laguerre weight. Finally, in Section 5, we work with a class of weight functions defined by w(x∕q) = Aw(x)∕(x2 + ax + b), which is a natural extension of the generalized Stieltjes-Wigert and q-Laguerre weights. We explore the properties of the subleading coefficient and of the recurrence coefficients. In particular, we derive the second- and third-order nonlinear difference equations and a system of difference equations for the subleading coefficient 𝛿 n .
2
LADDER O PERATO RS
Ladder operators are used by many authors to obtain nonlinear discrete equations for the recurrence coefficients of orthogonal polynomials. In this section, we shall recall the method following mainly Chen and Ismail9 and illustrate it on the examples of the classical Stieltjes-Wigert and q-Laguerre polynomials from previous studies.8,9,19-21 Let u be defined by u(x) = −
Dq−1 w(x) , w(x)
(12)
where Dq is the q-analogue of the difference operator, { (Dq 𝑓 )(x) =
𝑓 (x)−𝑓 (qx) x(1−q) ′
𝑓 (0)
if x ≠ 0, if x = 0.
9
Lemma 2.1. (Chen and Ismail ) Let ∞
An (x) = 𝛾n2
∫0
u(qx) − u(𝑦) Pn (𝑦)Pn (𝑦∕q)w(𝑦)d𝑦, qx − 𝑦
(13)
4
CHEN ET AL. ∞ 2 Bn (x) = 𝛾n−1
∫0
u(qx) − u(𝑦) Pn (𝑦)Pn−1 (𝑦∕q)w(𝑦)d𝑦. qx − 𝑦
(14)
Then the following lowering relation holds: Dq Pn (x) = a2n An Pn−1 (x) − Bn Pn (x).
(15)
Moreover, An (x) and Bn (x) satisfy the following equations: Bn+1 (x) + Bn (x) = (x − bn )An (x) + x(q − 1)
n ∑ A𝑗 (x) − u(qx),
(qS1 )
𝑗=0
a2n+1 An+1 (x) − a2n An−1 (x) = 1 + (x − bn )Bn+1 (x) − (qx − bn )Bn (x).
(qS2 )
Equations (qS1 ) and (qS2 ) will be referred to as the supplementary conditions. From (qS1 ) and (qS2 ), one can get another identity, “the sum rule,” which is often helpful: a2n An (x)An−1 (x) = Bn (x)2 + u(qx)Bn (x) + (1 + (1 − q)xBn (x))
n−1 ∑ A𝑗 (x), 𝑗=0
(qS1′ )
see Chen and Griffin12 for the proof. By eliminating Pn−1 (x) between (3) and (15), we get the following raising equation: Dq Pn−1 (x) = ((x − bn−1 )An−1 − Bn−1 )Pn−1 (x) − An−1 Pn (x).
(17)
The lowering and raising equations allow us to obtain a second-order linear difference equation for Pn (x). Indeed, let us set L1,n = Bn + Dq ,
(18)
L2,n = (x − bn−1 )An−1 − Bn−1 − Dq .
(19)
L1,n Pn = a2n An Pn−1 ,
(20)
L2,n Pn−1 = An−1 Pn .
(21)
Then
Now, by combining 2 equations above, we get a second-order q-difference equation L2,n (
1 L1,n Pn ) = a2n An−1 Pn , An
(22)
which is of the form D2q Pn (x) + Rn (x)Dq Pn (x) + Sn (x)Pn (x) = 0
(23)
with Rn (x) = Bn (qx) −
Dq An (x) An (qx) + (Bn−1 (x) − (x − bn−1 )An−1 (x)), An (x) An (x)
Sn (x) = a2n An (qx)An−1 (x) + Dq Bn (x) −
Bn (x) Dq An (x) An (x)
An (qx) + Bn (x) (Bn−1 (x) − (x − bn−1 )An−1 (x)). An (x)
(24)
(25)
CHEN ET AL.
5
In addition, replacing (x − bn−1 )An−1 − Bn−1 by Bn (x) − x(q − 1) L2,n = Bn (x) − x(q − 1)
∑n−1 𝑗=0
A𝑗 (x) + u(qx) using (qS1 ), we get
n−1 ∑ A𝑗 (x) + u(qx) − Dq .
(26)
𝑗=0
Example 1. The Stieltjes-Wigert weight is given by Askey19 (see also Boelen and Van Assche8 ) w(x) =
x𝛼 , q (−x; q)∞ (− x ; q)∞
x > 0,
0 < q < 1.
When 𝛼 = 0, it coincides with the weight given in Koekoek et al.22 , p544 We have ( ) q 1 q1−𝛼 u(x) = − 2 . 1−q x x
(27)
(28)
Note that if 𝛼 = 12 , then the function u(x) above is the same as for the weight w(x) = c exp((ln x)2 ∕(2 ln q)) considered 9 Moreover, when 𝛼 = 0, then the function u(x) above is the same as for the weight w(x) = in Chen ) (( and Ismail. )2 x ln √q ∕(2 ln q) considered in Ismail and Simeonov.21 Next, c exp u(qx) − u(𝑦) q−𝛼 u(𝑦) =− + , qx − 𝑦 qx (1 − q)x2 𝑦 An (x) =
Rn , x2
Bn (x) =
rn 1 − qn 1 − , 1−q x x2
(29)
(30)
where ∞ 𝛾n2 w(𝑦) Pn (𝑦)Pn (𝑦∕q) d𝑦, (1 − q)q𝛼 ∫0 𝑦 2 ∞ 𝛾n−1 w(𝑦) rn = Pn (𝑦)Pn−1 (𝑦∕q) d𝑦. (1 − q)q𝛼 ∫0 𝑦
Rn =
(31)
From the supplementary conditions (qS1 ) and (qS2 ), we get n ∑ qn (q + 1) − 1 = Rn + (q − 1) R𝑗 , 1−q 𝑗=0
(1 − q)(rn + rn+1 ) + (1 − q)Rn bn =
1 , q𝛼
(32)
(33)
qn bn − qrn + rn+1 = 0,
(34)
bn (rn − rn+1 ) = a2n+1 Rn+1 − a2n Rn−1 ,
(35)
1 where R0 = 1−q from Equation 32 by letting n = 0. A difference equation satisfied by Rn is found by subtracting Equation 32 with n − 1 from (32):
qRn − Rn−1 = −(1 + q)qn−1 .
(36)
The unique solution is Rn =
qn . 1−q
Using (33) together with (1 − q)Rn = qn and (34), one finds
(37)
6
CHEN ET AL.
1 . q𝛼
(38)
1 − q−n . (1 − q)q𝛼
(39)
rn − qrn+1 = The unique solution with r0 = 0 is given by rn = This, together with (34), immediately gives
bn = q−n−𝛼 (q−n−1 + q−n − 1).
(40)
Multiplying (35) by Rn and replacing bn Rn by (33), we find the difference equation ( ) rn+1 rn 2 2 rn+1 − 𝛼 − rn − 𝛼 = a2n+1 Rn+1 Rn − a2n Rn Rn−1 . q (1 − q) q (1 − q)
(41)
The solution with the initial conditions r0 = a20 = 0 is given by rn2 −
rn = a2n Rn Rn−1 . q𝛼 (1 − q)
(42)
This expresses a2n in terms of rn and Rn , a2n
rn = Rn Rn−1
(
1 rn − 𝛼 q (1 − q)
) = (q−4n − q−3n )q1−2𝛼 .
(43)
Finally, as bn = 𝛿 n − 𝛿 n+1 , from (40), we have 𝛿n =
q1−2n−𝛼 (qn − 1) . 1−q
(44)
Example 2. (Chen and Ismail9 ) The classical q-Laguerre weight is defined on R+ by w(x) =
x𝛼 , (−x; q)∞
where 𝛼 > 0, 0 < q < 1. It is easy to verify that q u(x) = 1−q
(
1 − q−𝛼 q−𝛼 + x x+q
) .
Then An (x) =
qn Rn − , x (1 − q)(x + 1)
Bn (x) =
rn qn−1 𝛿n − , x x+1
where Rn =
rn =
𝛾n2 (q−𝛼 − 1) ∞ w(𝑦) Pn (𝑦)Pn (𝑦∕q) d𝑦, ∫0 1−q 𝑦
2 (q−𝛼 − 1) 𝛾n−1
1−q
∞
∫0
Pn (𝑦)Pn−1 (𝑦∕q)
w(𝑦) d𝑦. 𝑦
By substituting these expressions into (qS1 ) and (qS2 ), we get that the recurrence coefficients bn and a2n are given by bn = q−2n−1−𝛼 (1 + q − qn+1 − qn+𝛼+1 ),
a2n = q−4n−2𝛼+1 (1 − qn )(1 − qn+𝛼 ),
which coincides with Koekoek et al.22 From bn = 𝛿 n − 𝛿 n+1 , we get
CHEN ET AL.
7
𝛿n =
q−𝛼−2n+1 (qn − 1)(q𝛼+n − 1) . q−1
For the weight given in Christiansen20 w(x) =
(−q∕x; q)∞ xc−1 , (−q𝛼+1−c x; q)∞ (−q−𝛼+c ∕x; q)∞ x𝛼 (−x;q)∞
the function u(x) is the same as for the weight w(x) =
in literature.9,21,22
For the weight given in Chihara23 (see also Boelen and Van Assche8 ) ( ) −𝑝 k 2 2 w(x) = √ ; q √ exp(−k ln x), qx 𝜋 ∞ we have q u(x) = 1−q
(
(45)
q 1 − √ x x( qx + 𝑝)
(46)
) .
(47)
The recurrence coefficients23 are as follows: 3
bn = q−n− 2 (−𝑝 − q + q−n+1 + q−n ),
a2n = q−4n (1 − qn )(1 − 𝑝qn−1 ).
As bn = 𝛿 n − 𝛿 n+1 , we have 1
q− 2 −2n (qn − 1)(𝑝qn − q) 𝛿n = . q−1 When p = q𝛼+1 and 𝛼 = 12 , q u(x) = 1−q which coincides with the case w(x) = Stieltjes-Wigert weight
x𝛼 (−x;q)∞
(
) √ q 1 − , x x(x + q)
(48)
with 𝛼 = 12 . When p = 0, the function u(x) is the same as for the classical w(x) = c exp((ln x)2 ∕(2 ln q)).
3
THE G ENERALIZED ST IELTJES-WIGERT WEIGHT
The generalized Stieltjes-Wigert weight function that is given in Boelen and Van Assche8 is defined by w(x) =
(−x2 ; q2 )
x𝛼 , 2 2 2 ∞ (−q ∕x ; q )∞
x ∈ [0, ∞),
0 < q < 1.
(49)
Following Boelen and Van Assche,8 we have w(x∕q) q2−𝛼 = 2 , w(x) x ( ) q 1 q2−𝛼 u(x) = − 3 . 1−q x x
(50)
(51)
The functions An and Bn , defined by (13) and (14) in the lowering ladder operator relation, are given8 by the following formulas: qn Tn + , (52) An (x) = 3 (1 − q)x (1 − q)x2
8
CHEN ET AL.
Bn (x) = −
1 − qn tn rn + , + (1 − q)x (1 − q)x2 (1 − q)x3
(53)
where ∞
Tn = 𝛾n2 q−𝛼
Pn (𝑦)Pn (𝑦∕q)
∫0
w(𝑦) d𝑦, 𝑦
∞ 2 q1−𝛼 rn = 𝛾n−1
∫0 ∫0
w(𝑦) d𝑦, 𝑦2
(55)
w(𝑦) d𝑦. 𝑦
(56)
Pn (𝑦)Pn−1 (𝑦∕q)
∞ 2 tn = 𝛾n−1 q−𝛼
Pn (𝑦)Pn−1 (𝑦∕q)
(54)
Lemma 3.1. The following relations hold8 true: Tn = qn−1 (𝛿n − q𝛿n+1 ),
(57)
rn = qn−1 (1 − q)𝛿n ,
(58)
( ) a2n = q1−2n q−𝛼 (1 − qn ) + qn tn ,
(59)
a2n =
(q − 1)𝛿n q1−2n−𝛼 , 𝛿n−1 − q𝛿n+1
(60)
a2n Tn Tn−1 = tn (tn − q−𝛼 ).
(61)
Let us examine additional equations that can be obtained from (qS2′ ). Equating the right-hand sides of (59) and (60), we obtain that tn can be expressed in terms of 𝛿 n as follows: ) ( (q − 1)𝛿n + qn − 1 . (62) tn = q−𝛼−n 𝛿n−1 − q𝛿n+1 Substituting An (x), Bn (x) from (52), (53) and u(x) from (51) into Equation (qS2′ ), we get the following equations: 𝛼
𝛼
qrn (2tn q − 1) − q
(a2n qn (qTn−1
n−1 ∑ + Tn ) + (q − 1)q T𝑗 tn ) = 0,
(63)
𝑗=0
a2n q𝛼+2n = q(−tn q𝛼+n − rn2 q𝛼 + (q − 1)rn q𝛼
n−1 ∑ T𝑗 + qn − 1),
(64)
𝑗=0 n−1 ∑ T𝑗 = 𝑗=0
rn . q−1
(65)
Using (58) to replace rn from (65), we get n−1 ∑ T𝑗 = −qn−1 𝛿n ,
(66)
𝑗=0
which coincides with the result when we take the sum over n for Tn given by (57). ∑ Substituting a2n from (59) and n−1 𝑗=0 T𝑗 from Equation 65 into (63), we obtain the following equation relating tn , rn , and Tn : qn rn (tn q𝛼 − 1) − (qTn−1 + Tn )(tn q𝛼+n − qn + 1) = 0. Substituting rn given by (58) and Tn given by (57) into this equation, we get (62). If we use (65) to replace (64), we can also obtain (59), which expresses a2n in terms of tn .
∑n−1 𝑗=0
T𝑗 from
CHEN ET AL.
3.1
9
Nonlinear difference equations for the subleading coefficient
We use expressions in the previous section to prove the following statement. Theorem 3.2. Let Pn = xn + 𝛿 n xn−1 + … be monic orthogonal polynomials for the generalized Stieltjes-Wigert weight (49) satisfying the three-term recurrence relation (3). Then the following statements hold true: (i) The sequence 𝛿 n with 𝛿 −1 = 𝛿 0 = 0, 𝛿 1 = −b0 satisfies the third-order difference equation given by 1 3 3 1 ∑ ∑∑∑ q r 𝑝 s c𝑝,q,r,s 𝛿n−1 𝛿n 𝛿n+1 𝛿n+2 = 0,
𝑝 + q + r + s ≤ 4,
(67)
𝑝=0 q=0 r=0 s=0
with 20 nonzero coefficients cp,q,r,s c0,0,1,1 = −q2 − q3 + qn+3 ,
c0,0,2,0 = q − q2 ,
c0,1,2,1 = −q𝛼+2n+2 − q𝛼+2n+3 ,
c0,1,1,0 = q + q2 − qn+2 ,
c0,2,1,1 = q𝛼+2n+2 ,
c0,2,0,0 = −q + q2 ,
c1,0,0,1 = −qn+2 + q2 + q, c1,1,0,0 = −1 − q + q1+n , c1,2,0,1 = −q𝛼+2n+1 ,
c0,0,3,1 = q𝛼+2n+3 ,
c0,1,0,1 = q2 − q3 ,
c0,1,3,0 = −q𝛼+2n+2 ,
c0,2,2,0 = q𝛼+2n+1 + q𝛼+2n+2 ,
c1,0,1,0 = −1 + q,
c0,3,1,0 = −q𝛼+2n+1 ,
c1,0,2,1 = −q𝛼+2n+2 ,
c1,1,1,1 = q𝛼+2n+1 + q𝛼+2n+2 ,
c1,2,1,0 = −q𝛼+2n − q𝛼+2n+1 ,
c1,1,2,0 = q𝛼+2n+1 ,
c1,3,0,0 = q𝛼+2n ;
(ii) The sequence 𝛿 n with 𝛿 −1 = 𝛿 0 = 0, 𝛿 1 = −b0 satisfies the second-order difference equation given by 2 3 2 ∑ ∑∑ 𝑝=0 q=0 r=0
𝑝 r c𝑝,q,r 𝛿n−1 𝛿n 𝛿n+1 = 0, q
𝑝 + q + r ≤ 4,
(68)
with 13 nonzero coefficients cp,q,r c0,0,2 = qn+3 − q3 ,
c0,1,1 = −qn+2 + qn+3 − 2q3 + 2q2 ,
c0,2,2 = q𝛼+2n+2 − q𝛼+2n+3 ,
c0,3,1 = q𝛼+2n+2 − q𝛼+2n+1 ,
c1,1,0 = qn+1 − qn+2 + 2q2 − 2q,
c1,0,1 = 2q2 − 2qn+2 ,
c1,1,2 = q𝛼+2n+2 − q𝛼+2n+1 ,
c1,2,1 = q𝛼+2n − 2q𝛼+2n+1 + q𝛼+2n+2 , c2,0,0 = qn+1 − q,
c0,2,0 = −q3 + 2q2 − q,
c1,3,0 = q𝛼+2n − q𝛼+2n+1 ,
c2,1,1 = q𝛼+2n − q𝛼+2n+1 ,
c2,2,0 = q𝛼+2n − q𝛼+2n−1 ;
(iii) Equations 67 and 68 are symmetric with respect to the change 𝛿 n → −𝛿 n .
Proof. We can get nonlinear difference equations by using (qS1 ), (qS2 ), (qS2′ ) and replacing all quantities there by the expressions in terms of the subleading coefficient 𝛿 n (after collecting the coefficients in x). Note that the second-order difference equation for 𝛿 n (68) is obtained from (61) and the description how to obtain the third-order Equation 67 is given in the proof of Theorem 3.3.
3.2
A nonlinear difference system
Theorem 3.3. Let Pn = xn + 𝛿 n xn−1 + … be monic orthogonal polynomials for the generalized Stieltjes-Wigert weight (49) satisfying the three-term recurrence relation (3). Then the sequence 𝛿 n satisfies a system of second-order difference equations 𝛿n+1 =
(1 + qn+𝛼 xn )𝛿n−1 − (q − 1)𝛿n (xn + q−𝛼 )2 , x x = . n−1 n+1 q + qn+1+𝛼 xn (qn+𝛼 xn + 1)2
We also have xn+1 = −q−𝛼 − xn + qn−1 (𝛿n − 𝛿n+1 )(q𝛿n+1 − 𝛿n ).
(69)
10
CHEN ET AL.
Proof. We use system (3.6a) to (3.7c) in Boelen and Van Assche,8 Equations 57, 58, 60, relation bn = 𝛿 n − 𝛿 n+1 and replace all quantities by the expressions in terms of 𝛿 n 's. We obtain a system of 3 equations tn + tn+1 = q−𝛼 + qn−1 (𝛿n − 𝛿n+1 )(q𝛿n+1 − 𝛿n ), ( ) 𝛿n (q𝛿n − 𝛿n−1 ) 𝛿n+1 (𝛿n+1 − q𝛿n+2 ) (tn − tn+1 )(𝛿n − 𝛿n+1 ) = a1 + , 𝛿n−1 − q𝛿n+1 𝛿n − q𝛿n+2 ( ) 𝛿n+1 𝛿n q(tn+1 − qtn ) + a2 (𝛿n − 𝛿n+1 )(q𝛿n+1 − 𝛿n ) − a3 + = 0, 𝛿n−1 − q𝛿n+1 q𝛿n+2 − 𝛿n where a1 = (q − 1)q−1−n−𝛼 , a2 = (q − 1)qn and a3 = −(q − 1)q1−n−𝛼 . Solving the first equation for tn+1 , tn+1 = q−𝛼 − tn + qn−1 (𝛿n − 𝛿n+1 )(q𝛿n+1 − 𝛿n ), we can derive 2 equations (either by eliminating tn or 𝛿 n+2 between the last 2 equations). The first one is the third-order difference equation for 𝛿 n (67), and second equation is 𝛿n+1 =
𝛿n−1 − qn 𝛿n−1 + qn+𝛼 tn 𝛿n−1 + 𝛿n − q𝛿n . q − qn+1 + q1+n+𝛼 tn
(70)
Since xn = tn − q−𝛼 in Boelen and Van Assche,8 we obtain a system of 2 equations (69). One of the equations in the system is the second-order difference equation for the subleading coefficient 𝛿 n of the orthogonal polynomial Pn (x), which also involves the solution xn of the discrete Painlevé III equation, and the second equation is the discrete Painlevé III equation itself (this equation was obtained in Boelen and Van Assche8 ).
The singularity confinement is used extensively for discrete integrable systems as a discrete analogue of the Painlevé property.18,24 The discrete Painlevé equations pass the singularity confinement test.18 The idea is that the singularity disappears after a finite number of iterations. A simple observation is that the same singularity gives rise to singularities for both xn+1 and 𝛿 n+1 in system (69). Therefore, it is interesting to analyze whether the singularity is confined to a finite number of iterations for (69). Assuming that xn−1 ≠ 0 and that xn gives rise to a singularity for the next iterate xn+1 and 𝛿 n+1 , we take xn = −
1 qn+𝛼
+ 𝜖.
(71)
Then the following expansions (in 𝜖) can be obtained: q−4n−4𝛼 (qn − 1)2 2q−3n−3𝛼 (qn − 1) q−2n−2𝛼 + +…, + xn−1 𝜖 xn−1 xn−1 𝜖 2 𝜖 = −q−2−n−𝛼 − 2 + … q (qn+2 − 1)2 xn−1 = +…, q4 (qn − 1)2 (1 − q)q−1−n−𝛼 𝛿n 𝛿n−1 = + +…, 𝜖 q (q − 1)2 q2n−3+2𝛼 xn−1 𝛿n 𝜖 𝛿 = n+ +…, q (qn − 1)2
xn+1 = xn+2 xn+3 𝛿n+1 𝛿n+2
and 𝛿 n+3 for 𝜖 = 0 is also finite. Thus, we see that the singularity is confined. If, in addition to (71), we have that the numerator is close to zero, ie, 𝛿n =
(1 + qn+𝛼 xn )𝛿n−1 + 𝜖, q−1
then using the system, we see that the singularity for 𝛿 n disappears in the first iteration and we have
CHEN ET AL.
11
𝛿n+1 = (1 − q)q−1−n−𝛼 , (q − 1 + qn+𝛼 𝛿n−1 )𝜖 +…, q(1 − q) 𝛿 = n−1 +…. q2
𝛿n+2 = 𝛿n+3
Thus, we see that the singularity is confined as well.
3.3
Expressions for bn
The recurrence coefficients bn in (8) and in (11) are defined via quadratic equations that are not very convenient for computations. If we want to calculate the polynomials explicitly, we need to use the three-term recurrence relation (3) and know the recurrence coefficients explicitly in terms of x0 and x1 (or z0 and z1 ). Since the coefficients bn are defined via quadratic equations, it is computationally not convenient to find (positive) roots in each step of the iteration. Therefore, a new algorithm is needed to calculate the coefficients bn efficiently. In this subsection, we present a new approach to compute these recurrence coefficients by using nonlinear difference equations found before. This approach can also be extended to other classes of semiclassical orthogonal polynomials. We use the nonlinear difference Equation 67 to compute the recurrence coefficient bn recursively (since it is linear in 𝛿 n+2 ). Substituting 𝛿 n = −b0 − … − bn−1 into (67) for consecutive n, we get that bn+1 is expressed in terms of b0 , … , bn . Note that 𝛿 −1 = 𝛿 0 = 0, 𝛿 1 = −b0 . Explicitly, the first few expressions are b1 = −
b0 (1 − 2q + q2+𝛼 b20 ) q(q1+𝛼 b20 − 1)
( b b2 = 02 2q
1+q+
(q − 1)2 1 + q − 2q1+𝛼 b20
−
, )
2q(q − 1) q1+𝛼 b20 − 1
.
It can be easily verified in any computer algebra system that these expressions solve the corresponding quadratic equation (1.8) in Boelen and Van Assche.8, Th.1.1
3.4
Formal asymptotic expansions
In this subsection, we find a few terms of the formal expansions of the subleading coefficient 𝛿 n for the Stieltjes-Wigert polynomials. Let us take the following ansatz: 𝛿n =
∞ ∑
ck qnk∕2 .
(72)
k=−2
Then, substituting this expansion into (68), we get successively q3∕4−𝛼∕2 , c−2 = √ q−1 c0 = q−1∕4−𝛼∕2
q3∕2 + 2q1∕2+𝛼 c2−1 − q𝛼 (1 + q)c2−1 . )3 (√ q−1
Coefficients c1 , c2 , c3 in (72) are cumbersome and we shall not present them here.
4
THE G ENERALIZED Q-LAGUERRE WEIGHT
In this section, the weight function is given by w(x) =
x𝛼 (−𝑝∕x2 ; q2 )∞ , 2 2 2 ∞ (−q ∕x ; q )∞
(−x2 ; q2 )
x ∈ [0, ∞),
𝑝 ∈ [0, q−𝛼 ),
0 < q < 1,
𝛼 ≥ 0.
(73)
12
CHEN ET AL.
We have w(x∕q) q2−𝛼 = 2 , w(x) x +𝑝 ( ) q q2−𝛼 1 u(x) = − . 1 − q x x(𝑝 + x2 )
(74) (75)
In Boelen and Van Assche,8 it is proved that the functions An and Bn , defined by (13) and (14) in the lowering ladder operator relation, are given by q2 Tn qn+2 + , 2 2 (1 − q)x(𝑝 + q x ) (1 − q)(𝑝 + q2 x2 )
(76)
1 − qn q2 rn q2 tn + + , (1 − q)x (1 − q)(𝑝 + q2 x2 ) (1 − q)x(𝑝 + q2 x2 )
(77)
An (x) = Bn (x) = − where
∞
Tn = 𝛾n2 q−𝛼
𝑦Pn (𝑦)Pn (𝑦∕q)
∫0
w(𝑦) d𝑦, 𝑝 + 𝑦2
∞ 2 q1−𝛼 rn = 𝛾n−1
∞ 2 q−𝛼 tn = 𝛾n−1
∫0
w(𝑦) d𝑦, 𝑝 + 𝑦2
(79)
w(𝑦) d𝑦. 𝑝 + 𝑦2
(80)
Pn (𝑦)Pn−1 (𝑦∕q)
∫0
Pn (𝑦)Pn−1 (𝑦∕q)
(78)
Note that An and Bn are expressed in terms of Tn , rn , tn , which have different definitions in each section. Lemma 4.1. The following relations hold true8 : Tn = qn−1 (𝛿n − q𝛿n+1 ),
(81)
rn = qn−1 (1 − q)𝛿n ,
(82)
( ) a2n = q1−2n −𝑝q−2 (1 − q2n ) + (1 − qn )(𝑝q−2 + q−𝛼 ) + qn tn ,
(83)
a2n =
(q − 1)𝛿n q1−2n−𝛼 , 𝛿n−1 − q𝛿n+1
tn+1 + tn + 𝑝q−2 (−1 + qn + qn+1 ) = −bn Tn + q−𝛼 .
(84) (85)
Since supplementary conditions (qS1 ) and (qS2 ) were used to derive the formulas above, let us examine equations that follow from (qS2′ ). First, equating the right-hand sides of (84) and (83), we obtain that tn can be expressed in terms of 𝛿 n as follows: ( n ) (q − 1)(q2 − 𝑝q𝛼+n ) (q − 1)𝛿n + tn = q−𝛼−n . (86) 𝛿n−1 − q𝛿n+1 q2 Substituting An (x), Bn (x) from (76), (77) and u(x) from (75) into Equation (qS2′ ), we get the following system: a2n Tn Tn−1 = 𝑝q2 (1 − qn )(𝑝q2 + q−𝛼 ) − 𝑝2 q4 (1 − q2n ) + tn (2𝑝qn−2 + tn − 𝑝q2 − q−𝛼 ), 𝛼
q
(a2n qn+1 (qTn−1
n
2
+ Tn ) + (q − 1)(𝑝q + q tn )
n−1 ∑
(87)
T𝑗 ) + rn (𝑝q𝛼 + q2 − 2𝑝q𝛼+n − 2tn q𝛼+2 ) = 0,
(88)
n−1 ∑ T𝑗 − 𝑝q𝛼+n+2 + 𝑝q𝛼+2n+2 − qn+4 + q4 = 0,
(89)
𝑗=0
rn2 q𝛼+4 − a2n q𝛼+2n+3 + tn q𝛼+n+4 − (q − 1)q𝛼+4 rn
𝑗=0
CHEN ET AL.
13 n−1 ∑ T𝑗 = 𝑗=0
rn . q−1
(90)
Equation 87 is the same as equation (4.9) in Boelen and Van Assche.8 By using (82) to replace rn in (90), we get n−1 ∑ T𝑗 = −qn−1 𝛿n ,
(91)
𝑗=0
which gives the same result as summing over n in (81). ∑ Substituting a2n from (83) and n−1 𝑗=0 T𝑗 from Equation 90 into (88), we obtain the following equation for tn , rn , and Tn : qn rn (𝑝q𝛼+n + tn q𝛼+2 − 𝑝q𝛼 − q2 ) − (qTn−1 + Tn )((qn − 1)(𝑝q𝛼+n − q2 ) + tn q𝛼+n+2 ) = 0. Substituting rn given by (82) and Tn given by (81) into this equation, we get the expression of tn in terms of 𝛿 n , which is in ∑ 2 agreement with (86). If we use (90) to replace n−1 𝑗=0 T𝑗 in (89), we can also obtain (83), which expresses an in terms of tn .
4.1
Nonlinear difference equations for the subleading coefficient
We use computations in the previous subsection to prove the following statement. Theorem 4.2. Let Pn = xn + 𝛿 n xn−1 + … be monic orthogonal polynomials for the generalized q-Laguerre weight (73) satisfying the three-term recurrence relation (3). Then the following statements hold true: (i) The sequence 𝛿 n with 𝛿 −1 = 𝛿 0 = 0, 𝛿 1 = −b0 satisfies the third-order difference equation given by 1 3 3 1 ∑ ∑∑∑ q r 𝑝 s d𝑝,q,r,s 𝛿n−1 𝛿n 𝛿n+1 𝛿n+2 = 0,
𝑝 + q + r + s ≤ 4,
(92)
𝑝=0 q=0 r=0 s=0
with 20 nonzero coefficients dp,q,r,s d0,0,1,1 = 𝑝q𝛼+n+1 + qn+3 − q3 − q2 ,
d0,0,3,1 = q𝛼+2n+3 ,
d0,0,2,0 = q − q2 ,
d0,1,1,0 = −𝑝q𝛼+n − qn+2 + q2 + q,
d0,1,0,1 = q2 − q3 ,
d0,1,2,1 = −q𝛼+2n+2 − q𝛼+2n+3 ,
d0,1,3,0 = −q𝛼+2n+2 ,
d0,2,0,0 = q2 − q,
d0,3,1,0 = −q𝛼+2n+1 ,
d1,0,0,1 = −𝑝q𝛼+n − qn+2 + q2 + q,
𝛼+2n+2
d1,0,2,1 = −q
d1,1,2,0 = q𝛼+2n+1 ,
,
𝛼+n−1
d1,1,0,0 = 𝑝q
d0,2,1,1 = q𝛼+2n+2 , n+1
+q
d1,2,0,1 = −q𝛼+2n+1 ,
d0,2,2,0 = q𝛼+2n+1 + q𝛼+2n+2 , d1,0,1,0 = −1 + q,
d1,1,1,1 = q𝛼+2n+1 + q𝛼+2n+2 ,
− q − 1,
d1,2,1,0 = −q𝛼+2n − q𝛼+2n+1 ,
d1,3,0,0 = q𝛼+2n ;
(ii) The sequence 𝛿 n with 𝛿 −1 = 𝛿 0 = 0, 𝛿 1 = −b0 satisfies the second-order difference equation given by 2 3 2 ∑ ∑∑ q r 𝑝 d𝑝,q,r 𝛿n−1 𝛿n 𝛿n+1 = 0,
𝑝 + q + r ≤ 4,
𝑝=0 q=0 r=0
with 13 nonzero coefficients dp,q,r d0,0,2 = −𝑝q𝛼+n+1 + 𝑝q𝛼+2n+1 − qn+3 + q3 , d0,1,1 = 𝑝q𝛼+n − 𝑝q𝛼+n+1 + qn+2 − qn+3 + 2q3 − 2q2 , d0,2,0 = q3 − 2q2 + q,
d0,2,2 = q𝛼+2n+3 − q𝛼+2n+2 ,
d0,3,1 = q𝛼+2n+1 − q𝛼+2n+2 ,
d1,0,1 = 2𝑝q𝛼+n − 2𝑝q𝛼+2n + 2qn+2 − 2q2 , d1,1,0 = −𝑝q𝛼+n−1 + 𝑝q𝛼+n − qn+1 + qn+2 − 2q2 + 2q, d1,2,1 = −q𝛼+2n + 2q𝛼+2n+1 − q𝛼+2n+2 ,
d1,3,0 = q𝛼+2n+1 − q𝛼+2n ,
d2,0,0 = −𝑝q𝛼+n−1 + 𝑝q𝛼+2n−1 − qn+1 + q, d2,1,1 = q𝛼+2n+1 − q𝛼+2n ,
d1,1,2 = q𝛼+2n+1 − q𝛼+2n+2 ,
d2,2,0 = q𝛼+2n−1 − q𝛼+2n ;
(iii) Equations 92 and 93 are symmetric with respect to 𝛿 n → −𝛿 n .
(93)
14
CHEN ET AL.
Proof. We can use Equations 81, 82, 84 in Lemma 4.1 and bn = 𝛿 n − 𝛿 n+1 . Substituting (81), (86), and bn = 𝛿 n − 𝛿 n+1 into (85), we get Equation 92. To obtain Equation 93, we substitute a2n , tn , and Tn in terms of 𝛿 n 's into Equation 87. Note that if p = 0, then Equations 92 and 93 reduce to (67) and (68).
4.2
A nonlinear difference system
Theorem 4.3. Let Pn = xn + 𝛿 n xn−1 + … be monic orthogonal polynomials for the generalized q-Laguerre weight (73) satisfying the three-term recurrence relation (3). Then the sequence 𝛿 n satisfies a system of second-order difference equations √ (q + 𝑝qn+𝛼∕2 zn )𝛿n−1 − (q − 1)q𝛿n , (94) 𝛿n+1 = √ q(q + 𝑝qn+𝛼∕2 zn )2 √ √ (zn + q2−𝛼 ∕𝑝)2 (zn + 𝑝q𝛼−2 )2 . (95) (zn zn−1 − 1)(zn zn+1 − 1) = √ (qn+𝛼∕2−1 𝑝zn + 1)2 We also have zn+1 = 𝑝−1∕2 q−1−𝛼∕2 (−q2 − 𝑝q𝛼 −
√
𝑝q1+𝛼∕2 zn + qn+1+𝛼 (𝛿n − 𝛿n+1 )(q𝛿n+1 − 𝛿n ).
(96)
Proof. Using (86), we can express 𝛿 n+1 in terms of tn , 𝛿 n−1 , and 𝛿 n . Then taking the change of variables zn =
tn + 𝑝qn−2 − 𝑝q2 − q−𝛼 , √ 𝑝q−𝛼−2
(97)
we immediately get (94). Equation 95 is given in Boelen and Van Assche.8 Substituting (81), (86), and bn = 𝛿 n − 𝛿 n+1 into (85), we get (96). It is shown in Boelen and Van Assche8 that Equation 95 is reduced to the second equation in (69) by taking √ xn = zn 𝑝q−1−𝛼∕2
(98)
and p → 0. We also have that by taking (98), Equation 94 is reduced to the first equation in (69). Now, we shall analyze the singularity confinement for systems (94) and (95). As in the previous section, assuming that zn = −
q−1−n−𝛼∕2 + 𝜖, √ 𝑝
with 𝜖 being small, we calculate the next iterates of both zn and 𝛿 n to show that the singularity is confined. We have zn+1 =
q1−4n−2𝛼 (qn − 1)2 (𝑝qn+𝛼 − q2 )2 +…, (√ ) 𝑝2 𝑝qn+𝛼∕2 + qzn−1 𝜖 2
zn+2 = −
q−1−n−𝛼∕2 𝜖 − 2+…, √ q 𝑝
(1 − q)q−n−𝛼∕2 𝛿n 𝛿n−1 +…, + √ q 𝑝𝜖 √ 2 2n−2+𝛼 ( 𝑝qn+𝛼∕2 + qz n−1 )𝛿n 𝜖 𝛿n 𝑝(q − 1) q = +…, + n 2 n+𝛼 2 2 q (q − 1) (𝑝q −q )
𝛿n+1 = 𝛿n+2
and both zn+3 and 𝛿 n+3 are finite when 𝜖 = 0. If in addition to (99), we have that √ (q + 𝑝qn+𝛼∕2 zn )𝛿n−1 𝛿n = + 𝜖, q(q − 1)
(99)
CHEN ET AL.
15
then the singularity for 𝛿 n disappears in the first iteration and we have 𝛿n+1 = 𝛿n+2 = 𝛿n+3 =
(1 − q)q−n−𝛼∕2 , √ 𝑝 √ (q2 − q + 𝑝qn+𝛼∕2 𝛿n−1 )𝜖 (q − 1)q2
+…,
𝛿n−1 +…. q2
Therefore, the singularity is confined as well.
4.3
Expressions for bn
As in the previous section for the generalized Stieltjes-Wigert polynomials, we can get expressions of bn+1 in terms of b0 , … , bn by using the third-order difference equation 92. Substituting 𝛿 n = −b0 − … − bn−1 into (92), we get an efficient way to calculate the sequence bn explicitly in terms of b0 . Using 𝛿 −1 = 𝛿 0 = 0, 𝛿 1 = −b0 , we get, for instance, b1 = −
b0 (1 − 2q + 𝑝q𝛼 + q𝛼+2 b20 ) (
b2 = b0
𝑝q𝛼 − q + q𝛼+2 b20
,
) ) ) ( ( (q − 1) q𝛼 b20 q2 + 𝑝 − q 1−q 1 , + ( + ) ) ) ( ( q q𝛼 b20 q2 + 𝑝 − q −2q𝛼+2 b20 q2 + 𝑝q + 𝑝 + 𝑝(q + 1)q2𝛼+1 b20 q2 + 𝑝 + (q + 1)q3
and so on.
4.4
Formal asymptotic expansions
As in the previous section, we can use the following ansatz for a formal expansion of the subleading coefficient: 𝛿n =
∞ ∑
dk qnk∕2 .
(100)
k=−2
Substitutung (100) into (93), we can get a few terms in the expansion: q3∕4−𝛼∕2 d−2 = √ , q−1 ( ) ) ( 𝛼 3 3 1 1 q− 2 − 4 d2−1 q𝛼+ 2 + q𝛼 𝑝 − d2−1 (q − 1)q − d2−1 q𝛼+ 2 + 𝑝q𝛼+ 2 + q5∕2 + q2 . d0 = )3 (√ ) (√ q−1 q+1 If p = 0, then d0 is the same as c0 in the previous section.
5
THE G ENERALIZED WEIGHT
In this section, we revisit papers25,26 (see also Boelen27 ) and obtain difference equations for the subleading coefficient of polynomials orthogonal with respect to a weight on R+ , which satisfies the property that the ratio w(x∕q)∕w(x) is a given rational function. We also assume that we are given the weight such that the moments are finite and all integrals in the proof exist (this might give certain restrictions on the parameters when we deal with a particular weight). We also show that the results in the previous 2 sections are the special cases of this general theory. In the following, we assume that the weight function satisfies ( ) x A = 2 w(x). (101) w q x + ax + b
16
CHEN ET AL.
For the Stieltjes-Wigert weight (49) A = q2−𝛼 , a = 0, b = 0 and for the q-Laguerre weight (73), one has A = q2−𝛼 , a = 0, b = p. We have the following statement. Theorem 5.1. (i) The recurrence coefficients a2n and bn in the three-term recurrence relation for monic orthogonal polynomials xPn (x) = Pn+1 (x) + bn Pn (x) + a2n Pn−1 (x), for the weight on R+ such that w
( ) x A = 2 w(x) q x + ax + b
can be expressed as follows: a2n = q1−n (𝑦n + Aq−n−2 ), q2n+2 A−1 (𝑦n+1 𝑦n − bAq−4 )b2n − qn−1 a(𝑦n+1 + 𝑦n + Aq−2 bq−2 )bn −(𝑦n+1 + 𝑦n + Aq−2 + bq−2 )2 = 0, where the function yn satisfies the second-order second-degree discrete equation (e2n − dn dn−1 )2 − cn cn−1 en (e2n + dn dn−1 ) − e2n (c2n dn−1 + c2n−1 dn ) = 0 with cn = qn−1 a, dn = q2n+2 A−1 (𝑦n+1 𝑦n − bAq−4 ), en =
(𝑦n + Aq−2 )(𝑦n + nq−2 ) . q1−n (𝑦n + Aq−n−2 )
(ii) Moreover, we have
( −n−2
tn = q
Tn = qn−1 (a + 𝛿n − q𝛿n+1 ),
(102)
rn = (1 − q)𝛿n qn−1 ,
(103)
) ( n )( ) A(q − 1)𝛿n n + q − 1 A − bq , a − q𝛿n+1 + 𝛿n−1
A(q − 1)𝛿n q−2n−1 , a − q𝛿n+1 + 𝛿n−1 ) ( (q − 1)𝛿n −n−2 𝑦n = Aq −1 . a − q𝛿n+1 + 𝛿n−1 a2n =
(104) (105) (106)
Proof. In the proof, we use the method from Filipuk and Smet26 (which, in turn, generalizes the proofs in Boelen and Assche25 and Boelen27 ). From (12), we have ( ) q A 1− 2 . u(x) = (1 − q)x (x + ax + b) Next, we use Aq u(qx) − u(𝑦) 1 = − u(𝑦) + 2 qx − 𝑦 qx (1 − q)(𝑦 + a𝑦 + b)(q2 x2 + aqx + b) A(𝑦 + a) + 2 (1 − q)x(𝑦 + a𝑦 + b)(q2 x2 + aqx + b)
(107)
CHEN ET AL.
17
to simplify the formulas ∞
∫0
u(𝑦)Pn (𝑦)Pn (𝑦∕q)w(𝑦)d𝑦 = 0,
∞
∫0
u(𝑦)Pn (𝑦)Pn−1 (𝑦∕q)w(𝑦)d𝑦 =
q 1 − qn . 2 1−q 𝛾n−1
(108)
(109)
Substituting (107) into (13), we get ∞ 𝛾n2 A 𝑦Pn (𝑦)Pn (𝑦∕q)w(𝑦) d𝑦 2 2 ∫ (1 − q)x(q x + aqx + b) 0 𝑦2 + a𝑦 + b ∞ 𝛾n2 A(qx + a) Pn (𝑦)Pn (𝑦∕q)w(𝑦) d𝑦. + 2 2 (1 − q)x(q x + aqx + b) ∫0 𝑦2 + a𝑦 + b
An =
Using (101), we can evaluate ∞
∫0
∞ Pn (𝑦)Pn (𝑦∕q)w(𝑦) Pn (𝑦)Pn (𝑦∕q)w(𝑦∕q) A−1 qn+1 . d𝑦 = d𝑦 = ∫0 A 𝑦2 + a𝑦 + b 𝛾n2
Similarly, ∞
∫0
∞ 𝑦Pn (𝑦)Pn (𝑦∕q)w(𝑦) 𝑦Pn (𝑦)Pn (𝑦∕q)w(𝑦∕q) d𝑦 d𝑦 = ∫0 A 𝑦2 + a𝑦 + b ∞
= q2 A−1
xPn (qx)Pn (x)w(x)dx
∫0 ∞
= q2 A−1
∫0
Pn (qx)(Pn+1 (x) + bn Pn (x) + a2n Pn−1 (x))w(x)dx,
where the recurrence relation (3) was used in the last step. To complete the evaluation of the integral above, we use the identity Pn (qx) = qn Pn (x) + 𝛿n (qn−1 − qn )Pn−1 (x) + · · · , the orthogonal relation and an = 𝛾 n−1 ∕𝛾 n to get the following result: ∞
q2 A−1
∫0
Pn (qx)(Pn+1 (x) + bn Pn (x) + a2n Pn−1 (x))w(x)dx ∞
(qn Pn (x) + 𝛿n (qn−1 − qn )Pn−1 (x) + · · · )(Pn+1 (x) + bn Pn (x) + a2n Pn−1 (x))w(x)dx ∫0 ( ) qn (𝛿n − 𝛿n+1 ) 𝛿n (qn−1 − qn )a2n 2 −1 =q A + 2 𝛾n2 𝛾n−1
= q2 A−1
=
q2 A−1 𝛾n2
qn−1 (𝛿n − q𝛿n+1 ).
Let Tn = qn−1 (a + 𝛿n − q𝛿n+1 ).
(110)
Then An (x) = Similarly, we can get
q2 (1 −
q)x(q2 x2
+ aqx + b)
Tn +
(1 −
qn+2 . + aqx + b)
q)(q2 x2
(111)
18
CHEN ET AL. 2 ∞ Aq 𝛾n−1 Pn (𝑦)Pn−1 (𝑦∕q)w(𝑦) 1 − qn 1 + d𝑦 2 2 1 − q x (1 − q)(q x + aqx + b) ∫0 𝑦2 + a𝑦 + b ( ) 𝑦 𝑦 2 ∞ Pn (𝑦)Pn−1 ( )w(𝑦) ∞ 𝑦Pn (𝑦)Pn−1 ( )w(𝑦) A 𝛾n−1 q q + a d𝑦 + d𝑦 ∫0 ∫0 (1 − q)x(q2 x2 + aqx + b) 𝑦2 + a𝑦 + b 𝑦2 + a𝑦 + b
Bn (x) = −
=−
(112)
q2 q2 1 − qn 1 + + r tn , n 1 − q x (1 − q)(q2 x2 + aqx + b) (1 − q)x(q2 x2 + aqx + b)
where ∞ 2 rn = q−1 𝛾n−1 A
∫0 ∞
2 A = q−1 𝛾n−1
∫0
Pn (𝑦)Pn−1 (𝑦∕q)w(𝑦) d𝑦 𝑦2 + a𝑦 + b Pn (𝑦)Pn−1 (𝑦∕q)w(𝑦∕q) d𝑦 A
(113)
∞ 2 = 𝛾n−1
∫0
Pn (qx)Pn−1 (x)w(x)dx
= (1 − q)𝛿n qn−1 , ∞ 2 tn = aq−2 𝛾n−1 A
∫0
∞ Pn (𝑦)Pn−1 (𝑦∕q)w(𝑦) 𝑦Pn (𝑦)Pn−1 (𝑦∕q)w(𝑦) −2 2 𝛾 A d𝑦 + q d𝑦 n−1 ∫0 𝑦2 + a𝑦 + b 𝑦2 + a𝑦 + b
∞ 2 = q−1 a𝛾n−1
∫0
∞ 2 Pn (qx)Pn−1 (x)w(x)dx + 𝛾n−1
∫0
(114)
xPn (qx)Pn−1 (x)w(x)dx.
To proceed further, we substitute An (x), Bn (x), and u(qx) into (qS1 ), multiply both sides by q−2 (1−q)x(q2 x2 +aqx+b), and equate the coefficients at the powers of x. We obtain the following equations: bq−2 (qn+1 − 1 + qn ) + tn+1 + tn = −bn Tn + Aq−2 , rn+1 + rn + a(qn + qn−1 − q−1 ) = Tn − bn qn + (q − 1)
n ∑ T𝑗 .
(115)
(116)
𝑗=0
Similarly, from (qS2 ), we have bn qn (1 − q) + rn+1 − qrn = 0,
(117)
bn qn−2 (1 − q)b − bn (tn+1 − tn ) = a2n+1 Tn+1 − a2n Tn−1 ,
(118)
tn+1 − qtn − bn rn+1 + bn rn + qn−1 (1 − q)abn = a2n+1 qn+1 − a2n qn−1 .
(119)
From (110) and (113), we get rn+1 − rn = (1 − q)(𝛿n+1 qn − 𝛿n qn−1 ) = (1 − q)(qn−1 a − Tn ). Inserting this expression into (119) gives tn+1 − qtn + (1 − q)bn Tn = a2n+1 qn+1 − a2n qn−1 .
(120)
Using (115) to replace bn Tn and multiplying both sides by qn , we get (1 − q)Aqn−2 + qn+1 tn+1 − qn tn + (1 − q)q−2 b(qn − q2n (1 + q)) = a2n+1 q2n+1 − a2n q2n−1 . Taking a telescopic sum, we obtain a2n q2n−1 = (1 − qn )(Aq−2 + bq−2 ) − (1 − q2n )bq−2 + qn tn . This gives an expression of a2n in terms of tn .
(121)
CHEN ET AL.
19
Next, multiplying (118) by Tn , using (115), (120), and (121), we obtain 2 − tn2 − (tn+1 − tn )(Aq−2 + bq−2 ) + qn−4 bA(1 − q) a2n+1 Tn+1 Tn − a2n Tn Tn−1 = tn+1
+ qn−4 b2 (1 − q) − b2 q2n−4 (1 − q2 ) + 2bqn−1 tn+1 − 2bqn−2 tn . Summing over n and taking into account that t0 = 0, a20 = 0, we get a2n Tn Tn−1 = tn2 − tn (Aq−2 + bq−2 ) + (bA + b2 )q−4 (1 − qn ) − b2 q−4 (1 − q2n ) + 2bqn−2 tn .
(122)
Replacing tn and tn+1 in (118) and using (121) yields a2n+1 (Tn+1 + bn qn ) − a2n (Tn−1 + bn qn−1 ) = Aq−n−3 (1 − q)bn . Substituting bn = 𝛿 n − 𝛿 n+1 and Tn from (110) gives a2n+1 q2n+1 (a + 𝛿n − q𝛿n+2 ) − a2n q2n−1 (a + 𝛿n−1 − q𝛿n+1 ) = Aq−2 (1 − q)bn . Taking the telescopic sum, we obtain A−1 a2n q2n+1 (a + 𝛿n−1 − q𝛿n+1 ) = (q − 1)𝛿n .
(123)
Using (110) and replacing 𝛿 n+1 = 𝛿 n − bn , we get qn bn = Tn − qn−1 (a + (1 − q)𝛿n ). Next, we replace 𝛿 n using (123) and again use 𝛿 n+1 = 𝛿 n − bn . We also express 𝛿 n in terms of Tn−1 and 𝛿 n−1 . Finally, we get bn qn (1 − q2n+1 A−1 a2n ) = Tn + A−1 q2n+2 a2n Tn−1 − qn−1 a.
(124)
Next, we multiply (124) by Tn and substitute bn Tn from (115) into it. This gives qn (1 − q2n+1 A−1 a2n )(Aq−2 − tn+1 − tn − bq−2 (qn+1 + qn − 1)) = Tn (Tn − qn−1 a) + A−1 q2n+2 a2n Tn−1 Tn .
(125)
In the last expression, we use (122) to replace a2n Tn−1 Tn and (121) to replace a2n . We also introduce a new variable 𝑦n = tn − Aq−2 − (1 − qn )bq−2 . Then Equation 125 can be written in new variables as Tn (Tn − qn−1 a) = q2n+2 A−1 (𝑦n+1 𝑦n − bAq−4 ).
(126)
q1−n (𝑦n + Aq−n−2 )Tn Tn−1 = (𝑦n + Aq−2 )(𝑦n + bq−2 ).
(127)
a2n = q1−n (𝑦n + Aq−n−2 ).
(128)
Equation 122 can be written as
From (121), we have Equations 126 and 127 can be used to eliminate Tn and Tn−1 to get a second-order second-degree nonlinear discrete equation for yn as follows. If we first rewrite these 2 equations as Tn2 − 𝜅n Tn − 𝜆n = 0,
(129)
Tn Tn−1 − 𝜇n = 0,
(130)
with 𝜅n = qn−1 a, 𝜆n = q2n+2 A−1 (𝑦n+1 𝑦n − bAq−4 ), 𝜇n =
(𝑦n + Aq−2 )(𝑦n + bq−2 ) , q1−n (𝑦n + Aq−n−2 )
20
CHEN ET AL.
this leads to 2 + 𝜅n 𝜇n Tn−1 − 𝜇n2 = 0. 𝜆n Tn−1
(131)
2 by computing the resultant. This gives Next, using this expression and Equation 129 with n − 1, we eliminate Tn−1 2 (𝜇n2 − 𝜆n 𝜆n−1 )2 − 𝜅n 𝜅n−1 𝜇n (𝜇n2 + 𝜆n 𝜆n−1 ) − 𝜇n2 (𝜅n2 𝜆n−1 + 𝜅n−1 𝜆n ) = 0.
(132)
On the other hand, substituting the expression of a2n in terms of yn from (128) into (124) and replacing Tn−1 from Equation 127, we get an expression for bn , yn , and Tn : Tn = −
1 (𝑦n+1 + 𝑦n + Aq−2 + bq−2 ). bn
(133)
Inserting Tn given by (133) into (126), we get a quadratic expression for bn in terms of yn and yn+1 : q2n+2 A−1 (𝑦n+1 𝑦n − bAq−4 )b2n − qn−1 a(𝑦n+1 + 𝑦n + Aq−2 bq−2 )bn − (𝑦n+1 + 𝑦n + Aq−2 + bq−2 )2 = 0.
(134)
As in previous section, we can also use (qS2′ ). Substituting An (x), Bn (x) from (111), (112) and u(x) from (107) into Equation (qS2′ ), we get Equation 122 and other 3 equations: aAqn − aA − 2abq2n + 2abqn − 2aqn+2 tn + aq2 tn + a2n qn+2 Tn + a2n qn+3 Tn−1 n−1 ∑ ( ) ( ) + qrn A − 2bqn + b − 2q2 tn + (q − 1)q T𝑗 bqn + q2 tn = 0,
(135)
𝑗=0
Aqn − A − bq2n + bqn − a2 q2n + a2 qn − qn+2 tn − q2 rn2 + a2n q2n+1 ( ) n−1 n−1 n−1 ∑ ∑ ∑ − aqn+1 T𝑗 + aqn+2 T𝑗 + qrn −2aqn + a + (q − 1)q T𝑗 = 0, 𝑗=0
𝑗=0 n−1 ∑
(136)
𝑗=0
T𝑗 =
𝑗=0
a(qn − 1) + qrn . (q − 1)q
(137)
Using (113) to replace rn from (137), we get n−1 ∑ a (qn − 1) − (q − 1)𝛿n qn T𝑗 = , (q − 1)q 𝑗=0
(138)
which agrees with summing the expression for Tn over n in (110). ∑n−1 Substituting a2n from (121) and 𝑗=0 T𝑗 from Equation 137 into (135), we get the following equation in term of tn , rn , and Tn : (
aqn − q2 Tn−1 − qTn
)( ) ) ) (( n ) ( ( ) q − 1 bqn − A + qn+2 tn + qn+1 rn −A + b qn − 1 + q2 tn = 0.
(139)
Substituting rn given by (113) and Tn given by (110) into this equation, we obtain tn in term of 𝛿 n : ( −n−2
tn = q
) ( n )( ) A(q − 1)𝛿n n + q − 1 A − bq . a − q𝛿n+1 + 𝛿n−1
From (123), we get an expression for a2n in terms of 𝛿 n :
(140)
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21
a2n =
A(q − 1)𝛿n q−2n−1 . a − q𝛿n+1 + 𝛿n−1
(141)
∑ Using Equation 141 to replace a2n from (121), we can also obtain tn in term of 𝛿 n . If we use (137) to replace n−1 𝑗=0 T𝑗 from (136), we obtain (121), which expresses a2n in terms of tn . Meanwhile, equating the right-hand sides of (128) and (141), we obtain that yn can be expressed in terms of 𝛿 n as follows: ( 𝑦n = Aq−n−2
) (q − 1)𝛿n −1 . a − q𝛿n+1 + 𝛿n−1
(142)
In particular, if a = 0, all 𝜅 n = 0, and we obtain 𝜇n2 = 𝜆n 𝜆n−1 from (132). This can be written as (𝑦n+1 𝑦n − bAq−4 )(𝑦n 𝑦n−1 − bAq−4 ) =
(𝑦n + Aq−2 )2 (𝑦n + bq−2 )2 , (A−1 qn+2 𝑦n + 1)2
which is a particular case of qPV . If we take a = b = 0, we get qPIII : 𝑦n+1 𝑦n−1 =
(𝑦n + Aq−2 )2 . (A−1 qn+2 𝑦n + 1)2
Remark 1. Equation 132 can be further simplified and written as follows: (𝜇n2 − 𝜆n 𝜆n−1 )2 = 𝜇n (𝜅n−1 𝜇n + 𝜅n 𝜆n−1 )(𝜅n 𝜇n + 𝜅n−1 𝜆n ). By setting Vn = qn+2 A−1 (𝑦n+1 𝑦n − bAq−4 ),
Un =
(𝑦n + Aq−2 )(𝑦n + bq−2 ) , (𝑦n + Aq−n−2 )
(143)
Equation 132 simplifies further to (Un2 − qVn Vn−1 )2 = a2 qn−2 Un (Un + qVn )(Un + Vn ).
(144)
However, the expressions of a2n and bn in terms of Un and Vn are cumbersome.
5.1
Nonlinear difference equations for the subleading coefficient
Theorem 5.2. The following equations hold for the subleading coefficient 𝛿 n of monic polynomials orthogonal with respect to the weight on R+ satisfying (101): (i) the third-order difference equation, which turns out to be an algebraic equation of total degree 4 in 𝛿 n−1 , 𝛿 n , 𝛿 n+1 , and 𝛿 n+2 given by 1 3 3 1 ∑ ∑∑∑ 𝑝=0 q=0 r=0 s=0
𝑝 r s h𝑝,q,r,s 𝛿n−1 𝛿n 𝛿n+1 𝛿n+2 = 0, q
𝑝 + q + r + s ≤ 4,
(145)
with 37 nonzero coefficients hp,q,r,s presented in Appendix A; (ii) the second-order difference equation, which turns out to be an algebraic equation of total degree 4 in 𝛿 n−1 , 𝛿 n , and 𝛿 n+1 , given by 2 3 2 ∑ ∑∑ 𝑝=0 q=0 r=0
𝑝 r h𝑝,q,r 𝛿n−1 𝛿n 𝛿n+1 = 0, q
with 23 nonzero coefficients hp,q,r presented in Appendix A.
𝑝 + q + r ≤ 4,
(146)
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CHEN ET AL.
Proof. Substituting tn from (140) and Tn from (110) into Equation 115 and using bn = 𝛿 n − 𝛿 n+1 , we can derive the third-order difference equation of 𝛿 n . This equation is not symmetric with respect to 𝛿 n → −𝛿 n unless a = 0. Substituting a2n given by (121) into (122), we get ( )( ) ) ( b qn − 1 bqn − A − q2 tn A − 2bqn + b + q4 tn2 (( ) )( ) − q3−2n Tn−1 Tn qn − 1 bqn − A + qn+2 tn = 0, Substituting tn and Tn from (140) and (110) into this equation, we deduce the second-order difference equation for 𝛿 n , which is also not symmetric unless a = 0. Let A = q2−𝛼 , a = 0, b = p. Then Equations 145 and 146 are consistent with (92) and (93). As in the previous sections, Equations 145 and 146 can be used to calculate a few terms in the formal asymptotic expansion of 𝛿 n and to calculate recursively the recurrence coefficient bn in terms of b0 . Since expressions are cumbersome and the calculations are analogous to the previous sections, we shall not present them here.
ACKNOWLEDGEMENTS G. Filipuk thanks W. Van Assche for useful discussions on initial conditions in Theorem 1.1 in Boelen and Van Assche.8 The support of Alexander von Humboldt Foundation is gratefully acknowledged. G. Filipuk also acknowledges the support of National Science Center (Poland) grant OPUS 2017/25/B/BST1/00931. H. M. Chen and Y. Chen would like to thank the Science and Technology Development Fund of the Macau SAR for generous support in providing FDCT 077/2012/A3, FDCT 130/2014/A3, and FDCT 023/2017/A1. We would also like to thank the University of Macau for generous support via MYRG 201400011 FST and MYRG 201400004 FST.
ORCID Hongmei Chen http://orcid.org/0000-0001-5115-7778 http://orcid.org/0000-0003-2623-5361 Galina Filipuk Yang Chen http://orcid.org/0000-0003-2762-7543
REFERENCES 1. Chihara TS. An Introduction to Orthogonal Polynomials. New York: Gordon and Breach; 1978. 2. Ismail ME. Classical and Quantum Orthogonal Polynomials in one Variable, Encyclopedia of Mathematics and Applications 98. Cambridge, UK: Cambridge University Press; 2005. 3. Szego G. Orthogonal Polynomials, Fourth Edition. Providence, RI: American Mathematical Society; 1975. 4. Filipuk G. Differential and difference equations for recurrence coefficients of orthogonal and multiple orthogonal polynomials. Habilitation thesis: University of Warsaw; 2016. 5. Filipuk G. The Painlevé equations and orthogonal polynomials. Electronic Notes Discrete Math. 2013;43:255-262. https://doi.org/10.1016/ j.endm.2013.07.042 6. Magnus AP. Painlevé-type differential equations for the recurrence coefficients of semi-classical orthogonal polynomials, in Proceedings of the Fourth International Symposium on Orthogonal Polynomials and their Applications (Evian-Les-Bains, 1992). J Comput Appl Math. 1995;57(1-2):215-237. https://doi.org/10.1016/0377-0427(93)E0247-J 7. Van Assche W. Discrete Painlevé equations for recurrence coefficients of orthogonal polynomials. In: Elaydi S, Cushing J, Lasser R, Papageorgiou V, Ruffing A, and Van Assche W eds. Difference Equations, Special Functions and Orthogonal Polynomials. Munich, Germany: World Scientific; 2007:687-725. 8. Boelen L, Van Assche W. Variations of Stieltjes-Wigert and q-Laguerre polynomials and their recurrence coefficients. J Approximation Theory. 2015;193:56-73. https://doi.org/10.1016/j.jat.2014.06.012 9. Chen Y, Ismail ME. Ladder operators for q-orthogonal polynomials. J Math Anal Appl. 2008;345:1-10. https://doi.org/10.1016/j.jmaa.2008. 03.031 10. Basor EL, Chen Y, Haq NS. Asymptotics of determinants of Hankel matrices via non-linear difference equations. J Approximation Theory. 2015;198:63-110. https://doi.org/10.1016/j.jat.2015.05.002 11. Chen Y. Its A. Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I. J Approximation Theory. 2010;162:270-297. https://doi.org/10.1016/j.jat.2009.05.005 12. Chen Y, Griffin J. Non linear difference equations arising from a deformation of the q-Laguerre weight. Indagationes Math. 2015;26(1):266-279. https://doi.org/10.1016/j.indag.2014.10.004 13. Ismail ME. Difference equations and quantized discriminants for q−orthogonal polynomials. Adv Appl Math. 2003;30(3):562-589. https:// doi.org/10.1016/S0196-8858(02)00547-X
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How to cite this article: Chen H, Filipuk G, Chen Y . Nonlinear difference equations arising from the generalized Stieltjes-Wigert and q-Laguerre weights. Math Meth Appl Sci. 2018;1–24. https://doi.org/10.1002/mma.4751
APPENDIX A The list of coefficients for Theorem 5.2: ( ( ) ) ) ( ( ) h0,0,0,0 = a2 A qn+1 − q − 1 + bqn+1 , h0,0,0,1 = −aq A qn+1 − q − 1 + bqn+1 , ) ( ( ) h0,0,1,0 = −a3 q2n+2 − a A qn+2 − q2 − 2q + 1 + bqn+2 , ( ) h0,0,1,1 = a2 q2n+3 + Aq2 qn+1 − q − 1 + bqn+3 , h0,0,2,0 = 2a2 q2n+3 − A(q − 1)q, h0,0,2,1 = −2aq2n+4 , h0,1,0,0
h0,0,3,0 = −aq2n+4 , h0,0,3,1 = q2n+5 , ( ( ) ) = a3 q2n+2 + a A qn+1 + q2 − 2q − 1 + bqn+1 ,
h0,1,0,1 = −a2 q2n+3 − A(q − 1)q2 ,
( ) h0,1,1,0 = −2a2 q2n+2 − 2a2 q2n+3 + A −qn+2 + q2 + q − bqn+2 , h0,1,2,0 = a(q + 3)q2n+3 ,
h0,1,2,1 = −q2n+4 − q2n+5 ,
h0,2,0,0 = 2a2 q2n+2 + A(q − 1)q,
h0,1,1,1 = a(2q + 1)q2n+3 ,
h0,1,3,0 = −q2n+4 ,
h0,2,0,1 = −aq2n+3 ,
h0,2,1,0 = −a(3q + 1)q2n+2 ,
h0,2,1,1 = q2n+4 , h0,2,2,0 = q2n+3 + q2n+4 , h0,3,0,0 = aq2n+2 , h0,3,1,0 = −q2n+3 , ( ( ) ( ) ) h1,0,0,0 = a A qn+1 − q − 1 + bqn+1 , h1,0,0,1 = A −qn+2 + q2 + q − bqn+2 , h1,0,1,0 = A(q − 1) − a2 q2n+2 , h1,0,1,1 = aq2n+3 , h1,0,2,0 = aq2n+3 , ( ) h1,1,0,0 = a2 q2n+2 + A qn+1 − q − 1 + bqn+1 , h1,1,0,1 = −aq2n+3 , h1,1,1,0 = −a(q + 2)q2n+2 , h1,2,0,1 = −q2n+3 , h0,0,0 = a
2
(
h1,1,1,1 = q2n+3 + q2n+4 ,
h1,2,1,0 = −q2n+2 − q2n+3 ,
n−1 )∑ bq − A qi , n
i=0
(
h1,1,2,0 = q2n+3 ,
h1,0,2,1 = −q2n+4 , h1,2,0,0 = 2aq2n+2 ,
h1,3,0,0 = q2n+2 ; n
h0,0,1 = 2aq A − bq
n−1 )∑ i=0
q, i
n+2
h0,0,2 = (bq
2
− Aq )
n−1 ∑ i=0
qi ,
24
CHEN ET AL.
( ( ) ) h0,1,0 = a A qn − 2 + bqn − a3 q2n , h0,1,2 = −aq2n+2 , h0,2,2 = q2n+3 ,
) ( ) ( h0,1,1 = qn+1 2a2 qn − b − Aq qn − 2 ,
h0,2,0 = (a2 q2n + A)(q − 1),
h0,3,0 = aq2n+1 ,
h0,2,1 = −a(2q − 1)q2n+1 ,
h0,3,1 = −q2n+2 ,
n−1 ( )∑ h1,0,0 = 2a bqn − A qi , i=0
h1,0,1
n−1 ∑ = (2Aq − 2bqn+1 ) qi ,
h1,1,0
(
)
) ( = A qn − 2 − qn 2a2 qn − b ,
h1,1,1 = 3aq2n+1 ,
i=0 2n+2
h1,1,2 = −q
,
h2,0,0 = (bqn − A)
h1,2,0 = a(q − 2)q2n , n−1 ∑ i=0
qi ,
h2,1,0 = −aq2n ,
h1,2,1 = −(q − 1)q2n+1 , h2,1,1 = q2n+1 ,
h1,3,0 = q2n+1 ,
h2,2,0 = −q2n .
