Ismail M., Koelink E. (editors) Theory and Applications of Special Functions

Подождите немного. Документ загружается.

Painleve' Equations and Associated Polynomials

Roots of

Rg(z)

Roots of

R.5 (z)

Roots of

R7(z)

Roots of

R4(z)

Roots of

Rs(z)

Figure

Roots of the Okamoto polynomials

defined by

(4.10)

150

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

which are defined in Theorem 4.9 and discussed in 54.4. Noumi and Ya-

mada (Noumi and Yamada, 1999) expressed both the generalized Her-

mite polynomials and the generalized Okamoto polynomials in terms of

Schur functions related to the so-called modified Kadomtsev-Petviashvili

(mKP) hierarchy. Kajiwara and Ohta (Kajiwara and Ohta, 1998) also

expressed rational solutions of

PIv

in terms of Schur functions by ex-

pressing the solutions in the form of determinants. Further Noumi and

Yamada (Noumi and Yamada, 1999) obtained their results on ratio-

nal solutions of PIv by considering the symmetric representation of PIv

given by the system

where p1, p2 and p3 are arbitrary constants, with pl

and

the constraint

cpl+

-22. Then eliminating ~~(2) and cp3(z),

w(z)

cpl(z) satisfies PIV with (a, P)

(p3

pa, -2&), which was

first derived by Bureau (Bureau, 1992)

see also (Adler, 1994; Noumi

and Yamada, 1998a; Schiff, 1995; Veselov and Shabat, 1993).

First we discuss the generalized Hermite polynomials Hm,,(z).

Theorem

4.7.

Suppose that

Hm,,(z)

satisfies the recurrence relations

2mHrn+1,nHm-l7n

Hm,nH;,,

(H;,,)

2m~;,,,

(4.18a)

2nHm,n+lHm,n-l

-Hm,nH;,

(H;,~)~

2n~;,,,

(4.18b)

with

Ho,o

Hi,o

Ho,i

H1,i

22,

(4.18~)

and

m, n

then

is a solution of

PIv,

respectively for the parameters

a:!,

-(m

1),

&),

-2m2,

(4.21)

a:!!

,B:!i

-2n2. (4.22)

Painleve' Equations and Associated Polynomials

151

Remarks

4.8.

1. The rational solutions of PIv defined by (4.19) and (4.20) include

all the solutions in the "-l/zV and "-22" hierarchies,

is easily

verified by comparing the parameters in (4.21) and (4.22) with

those in (4.5a) and (4.5b).

Further they are the set of rational

solutions of

PIv

with parameter values given by (4.1).

2. Each generalized Hermite polynomial Hm,,(z) is a polynomial of

degree

with integer coefficients (Noumi and Yamada, 1999).

In fact

Hmjn

($x)

is a monic polynomial in

of degree

with

integer coefficients.

3. The polynomials H,,,(z) possess the symmetry

Hm,,(iz)

imn Hn,,(z), where

is the degree of Hm,,(z).

4. Hn,l(z)

Hn(z) and H1,,(z)

PnHn(iz), where Hn(z) is the

usual Hermite polynomial defined by

(z)

1)" exp(z2) {exp(--z2))

(4.23)

Some generalized Hermite polynomials Hm,,(z) are given in Appendix

A. Plots of the locations of the roots of the polynomials and Hm,7(z),

H7,n(z) for 4

6 and 4

6, are given in Figure

These plots,

which are invariant under reflections in the real and imaginary z-axes,

take the form of

"rectangles", though these are only approximate

rectangles

can be seen by looking at the actual values of the roots.

4.4

Generalized Okamoto polynomials

In this section we discuss the generalized Okamoto polynomials Pm,,(z)

which were introduced by Noumi and Yamada (Noumi and Yamada,

1999) and are defined in Theorem 4.9 below. We have reindexed these

polynomials by setting Q,,, (z)

Pm-,,,

(z), i.e., Qm+,,, (z)

Pm,,

(z),

where Qm+n,n(z) is the polynomial defined Noumi and Yamada (Noumi

and Yamada, 1999), since we feel that Pm,,(z) is more natural, espe-

cially when one studies the plots of the locations of the roots for various

generalized Okamoto polynomials in Figure

Theorem

4.9.

Suppose that Pmln(z) satisfies the recurrence relations

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

Roots of

H4,7(z)

Roots of

H7,5(~)

Roots of

H5,7(z)

Roots of

H7,4(2)

Figure

Roots of generalized Hermite polynomials defined by

(4.18)

Painleve' Equations and Associated Polynomials

153

with

Po,o

P1,o

Po,l

PIJ

&z,

(4.24~)

then

(11)

2 d Pm+

(I1)

P("))

--z

{ln

(-) )

(4.26)

w (2;

am,n,

m,n

Pm,n

are solutions of

PIv,

respectively for the parameters

Remarks

4.10.

1. The rational solutions of

PIv

defined by (4.25) and (4.26) include

all the solutions in the "-$2" hierarchy,

is easily verified by

comparing the parameters in (4.27) and (4.28) with those in (4.5~).

Further they are the set of rational solutions of

PIv

with parameter

values given by (4.2).

2. Each polynomial Pm,,(z) is a polynomial of degree dm,,

rnn

with integer coefficients (Noumi and Yamada,

1999). Further Pm,,(z) is a monic polynomial in

z of degree

dm,, with integer coefficients.

3. The original Okamoto polynomials defined in Theorems 4.3 and 4.5

are respectively given by Qm(z)

Pmlo(z) and &(z)

Pm,l

(2).

4. The polynomials

Pmln(2)

possess the symmetry

Pm,n(iz)

exp($ridm,,) Pn,m(z), where dm,,

is the degree of Pm,,(z).

5. The hierarchies of rational solutions of

PIv

generated from the gen-

eralized Hermite polynomials Hm,n(z) defined in Theorem 4.7 and

the generalized Okamoto polynomials Pmln(z) defined in Theorem

4.9 are linked by the Schlesinger transformations

%d2]

(or

7d41)

and

R[~]

R[']R[~]

given by Fokas, Mugan and Ablowitz (Fokas et al.,

1988).

Some generalized Okamoto polynomials Pm,,(z) are given in Ap-

pendix

Plots of the locations of the roots of the polynomials and

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

Roots of

P4,7(z)

Roots of

P5,7(z)

Roots of

P6,7(z)

Roots of

P74(z)

Roots of

P7,5(~)

Roots of

P7,6(~)

Figure

Roots of generalized Okamoto polynomials defined by

(4.24)

Painleve' Equations and Associated Polynomials

155

Pm,7(~), P~,,(z) for 4

6 and

are given in Figure

The roots of the polynomial Pm,,(z) takes the form of

"rectangles"

with an "equilateral triangle," which have either

roots on

each of its sides. These are only approximate rectangles and equilateral

triangles as can be seen by looking at the actual values of the roots. The

triangles We remark that as for the Bi-Hermite polynomials above, the

plots are invariant under reflections in the real and imaginary z-axes.

Conclusions

An important, well-known property of classical orthogonal polynomi-

als, such as the Hermite, Laguerre or Legendre polynomials whose roots

all lie on the real line (cf. (Abramowitz and Stegun, 1972; Andrews et al.,

1999; Temme, 1996)), is that the roots of successive polynomials inter-

lace. Thus for a set of orthogonal polynomials p,(z), for

0,1,2,.

. .

if znlm and z,,,+l are two successive roots of p,(z), i.e.,

(z,,,)

and pn (zn,m+l)

0, then

9,-1

(<,-I)

and pn+l (&+I)

for some

and such that

zn,,

<,+i

zn,,+l. Further the deriva-

tives pL(z) and ~;+~(z) also have roots in the interval (zn,m,zn,m+l),

that is

(tn)

0 and

pL+l

(en+l)

for some

and such that

zn,m

tn,

tn+l

zn,m+l-

An interesting open question is whether there are analogous results

for the polynomials associated with rational and algebraic solutions of

the Painlev6 equations discussed here. Clearly there are notable differ-

ences since these special polynomials have complex roots whereas clas-

sical orthogonal polynomials p,(~), have real roots. The pattern of the

roots of the special polynomials associated with the Painlev6 equations

are highly symmetric and structured, suggesting that they have inter-

esting properties. A particularly interesting question is whether there

is any "interlacing of roots" analogous to that for classical orthogonal

polynomials. Clearly this warrants further analytical study as does an

investigation of the relative locations of the roots for the special polyno-

mials and their derivatives. We shall not pursue these questions further

here.

Acknowledgments

I wish to thank Mark Ablowitz, Frederick Cheyzak, Chris Cosgrove,

Andy Hone, Nalini Joshi, Arieh Iserles, Alexander Its, Elizabeth Mans-

field and Marta Mazzocco for their helpful comments and illuminating

discussions. This research arose whilst involved with the "Digital Library

of Mathematical Functions" project (see

http:

//dlmf

.nist

.gov/),

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

particular

thank Frank Olver and Dan Lozier for the opportunity to

participate in this project.

References

Ablowitz, M.

and Clarkson, P. A.

(1991).

Solitons, Nonlinear Evolution Equations

and Inverse Scattering, volume

149

of Lecture Notes in Mathematics. Cambridge

University Press, Cambridge.

Ablowitz, M.

and Satsuma,

(1978).

Solitons and rational solutions of nonlinear

evolution equations.

Math. Phys.,

19:2180-2186.

Abramowitz, M. and Stegun, I. A.

(1972).

Handbook of Mathematical Functions.

Dover, New York, tenth edition.

Adler,

and Moser,

(1978).

On a class of polynomials associated with the

Korteweg-de Vries equation. Commun. Math. Phys.,

61:l-30.

Adler, V.

(1994).

Nonlinear chains and Painlev6 equations. Physica,

D73:335-351.

Airault, H.

(1979).

Rational solutions of Painlev6 equations. Stud. Appl. Math.,

61:31-

53.

Airault, H., McKean,

P.,

and

Moser,

(1977).

Rational and elliptic solutions of

the KdV equation and related many-body problems. Commun. Pure Appl. Math.,

30:95-148.

Albrecht, D.

W.,

Mansfield,

L., and Milne, A.

(1996).

Algorithms for special

integrals of ordinary differential equations.

Phys. A: Math. Gen.,

29:973-991.

Andrews,

G.,

Askey, R., and Roy,

(1999).

Special Functions, volume

of Ency-

clopedia of Mathematics and its Applications. Cambridge University Press, Cam-

bridge.

Bassom, A. P., Clarkson, P.

A.,

and Hicks, A. C.

(1995).

Backlund transformations

and solution hierarchies for the fourth Painlevk equation. Stud. Appl. Math.,

95:l-

71.

Bureau, F.

(1992).

Differential equations with fixed critical points. In Painleve' tran-

scendent~ (Sainte-AdBle, PQ, 1990), volume

278

of NATO Adv. Sci. Inst. Ser.

Phys., pages

103-123.

Plenum, New York.

Charles, P.

(2002).

Painleve' analysis and the study of continuous and discrete

Painleve' equations. PhD thesis, Institute of Mathematics and Statistics, University

of Kent, UK.

Clarkson, P. A. and Kruskal, M. D.

(1989).

New similarity solutions of the Boussinesq

equation.

Math. Phys.,

30:2201-2213.

Clarkson, P. A. and Mansfield,

(2003).

The second Painlev6 equation, its hier-

archy and associated special polynomials. Nonlinearity,

16:Rl-R26.

Flaschka, H. and Newell, A. C.

(1980).

Monodromy-and spectrum preserving defor-

mations.

Commun. Math. Phys.,

76:65-116.

Fokas, A. S. and Ablowitz, M.

(1982).

On a unified approach to transformations

and elementary solutions of Painlev6 equations.

Math. Phys.,

23:2033-2042.

Fokas, A. S., Mugan, U., and Ablowitz, M.

(1988).

A method of linearisation for

Painlev6 equations: Painlevk IV, V. Physica,

D30:247-283.

Fokas, A. S. and Yortsos, Y. C.

(1981).

The transformation properties of the sixth

painlev6 equation and one-parameter families of solutions. Lett. Nuovo Cim.,

30:539-

544.

Painleve' Equations and Associated Polynomials

157

Fukutani, S., Okarnoto, K., and Umemura, H. (2000). Special polynomials and the

Hirota bilinear relations of the second and fourth Painlevk equations. Nagoya Math.

J., 159:179-200.

Gambier, B. (1910). Sur les kquations diffkrentielles du second ordre et du premeir

degre dont l'intkgrale gknkrale est

points critiques fixks. Acta Math., 33:l-55.

Gromak,

(1973). The solutions of Painlevk's third equation. Diff. Eqns., 9:1599-

1600.

Gromak, V.

(1975). Theory of Painlevk's equation. Diff. Eqns., 11:285-287.

Gromak, V.

(1976). Solutions of Painlevk's fifth equation. Diff. Eqns., 12:519-521.

Gromak, V.

(1978a). Algebraic solutions of the third Painlevk equation. Dokl. Akad.

Nauk BSSR, 23:499-502.

Gromak, V.

(197813). One-parameter systems of solutions of Painlevk's equations.

Diff. Eqns., 14:1510-1513.

Gromak, V.

(1987). Theory of the fourth painlevk equation. Diff. Eqns., 23:506-513.

Gromak, V.

(1999). Backlund transformations of Painlevk equations and their ap-

plications.

Conte, R., editor, The Painlevk Property, One Century Later, CRM

Series in Mathematical Physics, pages 687-734. Springer, New York.

Gromak, V.

(2001). Backlund transformations of the higher order Painlevk equa-

tions. In Coley, A., Levi,

D.,

Milson, R., Rogers, C., and Winternitz, P., editors,

Backlund and Darboux transformations. The geometry of solitons (Halifax, NS,

1999), volume 29 of CRM Proc. Lecture Notes, pages 3-28. Amer. Math. Soc.,

Providence, RI.

Gromak, V.

I.,

Laine,

I.,

and Shimomura, S. (2002). Painlevk Differential Equations

the Complex Plane, volume 28 of Studies in Mathematics. de Gruyter, Berlin,

New York.

Gromak,

and Lukashevich, N.

(1982). Special classes of solutions of Painlevk's

equations. Diff. Eqns., 18:317-326.

Iwasaki, K., Kajiwara, K., and Nakamura, T. (2002). Generating function associated

with the rational solutions of the Painlevk I1 equation. J. Phys. A: Math. Gen.,

35:L207-L211.

Iwasaki, K., Kimura, H., Shimomura, S., and Yoshida, M. (1991). From Gauss to

Painlevk: a Modern Theory of Special Functions, volume 16 of Aspects of Mathe-

matics E. Viewag, Braunschweig, Germany.

Jimbo, M. and Miwa, T. (1983). Solitons and infinite dimensional Lie algebras. Publ.

RIMS, Kyoto Univ., 19:943-1001.

Kajiwara, K. and Masuda,

(1999a). A generalization of determinant formulae

for the solutions of Painlevk I1 and XXXIV equations.

Phys. A: Math. Gen.,

32:3763-3778.

Kajiwara, K. and Masuda, T. (1999b). On the Umemura polynomials for the Painlevk

I11 equation. Phys. Lett., A260:462-467.

Kajiwara, K. and Ohta,

(1996). Determinantal structure of the rational solutions

for the painlevk I1 equation. J. Math. Phys., 37:4393-4704.

Kajiwara, K. and Ohta,

(1998). Determinant structure of the rational solutions for

the Painlevk

equation.

Phys. A: Math. Gen., 31:2431-2446.

Kametaka,

(1983). On poles of the rational solution of the Toda equation of

PainlevbII type. Proc. Japan Acad. Ser. A Math. Sci., 59:358-360.

THEORY AND APPLICATIONS OF SPECIAL FUNCTIONS

Kametaka, Y. (1985). On the irreducibility conjecture based on computer calculation

for YablonskiY-Vorobev polynomials which give a rational solution of the Toda

equation of Painlev6-11 type. Japan

Appl. Math., 2:241-246.

Kaneko, M. and Ochiai, H. (2002). On coefficients of Yablonskii-Vorob'ev polynomi-

als. Preprint, arXiv:math.QA/0205178.

Kirillov, A. N. and Taneda, M. (2002a). Generalized Umemura polynomials. Rocky

Mount.

Math., 32:691-702.

Kirillov, A.

and Taneda, M. (2002b). Generalized Umemura polynomials and

Hirota-Miwa equations. In Kashiwara, M. and Miwa, T., editors, MathPhys Odyssey,

2001,

volume 23 of Prog. Math. Phys., pages 313-331. Birkhauser-Boston, Boston,

MA.

Lukashevich, N. A. (1965). Elementary solutions of certain Painlev6 equations. Diff.

Eqns., 1:561-564.

Lukashevich, N. A. (1967a). On the theory of the third Painlev6 equation. Diff. Eqns.,

3:994-999.

Lukashevich, N. A. (196713). Theory of the fourth Painlev6 equation. Diff. Eqns.,

3:395-399.

Lukashevich, N. A. (1968). Solutions of the fifth equation of painlev6 equation. Diff.

Eqns., 4:732-735.

Lukashevich, N. A. (1971). The second painlevk equation. Diff. Eqns., 6:853-854.

Lukashevich, N. A. and Yablonskii, A. I. (1967). On

class of solutions of the sixth

painlevk equation. Diff. Eqns., 3:264-266.

Masuda, T. (2002). On a class of algebraic solutions to Painlev6 VI equation, its

determinant formula and coalescence cascade. preprint, arXiv:nlin.S1/0202044.

Masuda,

T.,

Ohta, Y., and Kajiwara, K. (2002). A determinant formula for a class of

rational solutions of Painlev6 V equation. Nagoya Math. J., 168:l-25.

Mazzocco, M. (2001). Rational solutions of the Painlev6 VI equation.

Phys. A:

Math. Gen., 34:2281-2294.

Milne, A. E., Clarkson, P. A., and Bassom, A. P. (1997). Backlund transformations

and solution hierarchies for the third Painlev6 equation. Stud. Appl. Math., 98:139-

194.

Murata, Y. (1985). Rational solutions of the second and the fourth Painlev6 equations.

Funkcial. Ekvac., 28:l-32.

Murata, Y. (1995). Classical solutions of the third Painlev6 equations. Nagoya Math.

J., 139:37-65.

Noumi, M. and Yamada, Y. (1998a). Affine Weyl groups, discrete dynamical systems

and Painlev6 equations. Commun. Math. Phys., 199:281-295.

Noumi, M. and Yamada, Y. (1998b). Umemura polynomials for the Painlev6 V equa-

tion. Phys. Lett., A247:65-69.

Noumi, M. and Yamada, Y. (1999). Symmetries in the fourth Painlev6 equation and

Okamoto polynomials. Nagoya Math.

J.,

153:53-86.

Noumi M., Okada

S.,

K. and H.,

(1998). Special polynomials msociated with

the Painlev6 equations. 11. In Saito, M.-H., Shimizu, Y., and Ueno,

R.,

editors,

Integrable Systems and Algebraic Geometry, pages 349-372. World Scientific, Sin-

gapore.

Ohyama, Y. (2001). On the third Painlev6 equations of type

d7.

Preprint, Department

of Pure and Applied Mathematics, Graduate School of Information Science and

Technology, Osaka University.