On the properties for modifications of classical orthogonal polynomials of discrete variables

ON THE PROPERTIES FOR MODIFICATIONS OF CLASSICAL ORTHOGONAL POLYNOMIALS OF DISCRETE VARIABLES. 1 R.  Alvarez-No darse, A.G.Garca and F. Marcellan Departamento de Ingeniera. Escuela Politecnica Superior. Universidad Carlos III de Madrid. Butarque 15, 28911,Leganes,Madrid. Key words and phrases: Meixner,Charlier and Kravchuk p olynomials, discrete measures, hyp ergeometric functions, asso ciated p olynomials. AMS (MOS) sub ject classication: 33A65 Abstract We consider a mo dication of moment functionals for some classical p olynomial s of a discrete variable by adding a mass p oint at x = 0. We obtain the resulting orthogonal p olynomial s, identify them as hyp ergeometric functions and derive the second order dierence equation which these p olynomial s satisfy. The corresp onding tridiagonal matrices and asso ciated p olynomials were also studied. x 1 Intro duction. The study of orthogonal p olynomials with resp ect to a mo dication of a linear functional in the linear space of p olynomials with real co ecients via the addition of one or two delta Dirac measures has b een p erformed by several authors. In particular, Chihara [5] has considered some prop erties of such p olynomials in terms of the lo cation of the mass p oint with resp ect to the supp ort of a p ositive measure. More recently Marcellan and Maroni [10] analyzed a more general situation for regular ( quasi-denite ) linear functionals, i.e., such that the principal submatrices of the corresp onding innite Hankel matrices asso ciated with the moment sequences are nonsingular. A sp ecial emphasis is given to the mo dications of classical linear functionals (Hermite, Laguerre, Jacobi and Bessel). Ko ornwinder [9] considered a system of p olynomials orthogonal with resp ect to the classical weigtht function for Jacobi p olynomials with two extra p oint masses added at x =  1 and x = 1. For generalized Laguerre p olynomials f L ;A n ( x ) g 1 n =0 that are orthogonal on [0 ; 1 ) with resp ect to the linear functional C on the linear space of p olynomials with real co ecients dened as < C ; P > = Z 1 0 P ( x ) x  e  x dx + AP (0) ;  >  1 ; A  0 ; Ko eko ek and Ko eko ek [8] found a dierential equation of the form A 1 X i =0 a i ( x ) y ( i ) ( x ) + xy 00 ( x ) + (  + 1  x ) y 0 ( x ) + ny = 0 ; where the co ecients a i ( x ) ; i 2 f 1 ; 2 ; 3 ; ::: g , are indep endent of n and a 0 dep ends on n but is indep endent of x . In the ab ove pap er, representation formulas for the new orthogonal 1 Octob er 29, 1996 1 p olynomial sequences, as well as the second order dierential equation that such p olynomials satisfy were deduced. In the op en problem section of the Pro ceedings of the Third International Symp osium on Orthogonal Polynomials and their Applications held in Erice (Italy), R. Askey raised the following question [1]: Consider the Meixner Polynomials M  ; n ( x ) , add or substract a mass point at x = 0 and nd the resulting polynomials. Identify them as hypergeometric functions and show that these polynomials satisfy a dierence equation in x . In [3] Bavinck and van Haeringen gave the solution to the problem of nding the second order dierence equation for generalized Meixner p olynomials, as well as the innite order dierence equation which these p olynomials satisfy. For generalized Charlier p olynomials Bavinck and Ko eko ek [4] found the corresp onding innite order dierence equation. In the pap er [2] we obtained the representation for such generalized Meixner p olynomials as an hyp ergeometric function 3 F 2 , as well as the corresp onding second order dierence equation. Now we generalize this result for the Kravchuk and Charlier p olynomials and continue the algebraic approach presented by Go doy, Marcellan, Salto, Zarzo ( see [7] ) in the framework of a more general theory based in the addition of a delta Dirac measure to a discrete semiclassical linear functional. We analyze the relation b etween tridiagonal matrices of the perturbed or generalized P A n ( x ) and classical P n ( x ) p olynomials, as well as the asso ciated p olynomials corresp onding to the sequence f P A n ( x ) g 1 n =0 . The structure of the pap er is as follows. In Section 2, we provide the basic prop erties of the classical orthogonal p olynomials of discrete variable which will b e needed, as well as the main data for the Meixner, Kravchuk and Charlier p olynomials. In Section 3 we deduce expressions of the generalized Meixner, Kravchuk and Charlier p olynomials and its rst dierence derivatives, as well as their representation as hyp ergeometric functions in the direction raised by Askey. In Section 4, we nd the second order dierence equation which these generalized p olynomials satisfy. In Section 5, from the three term recurrence relation (TTRR) of the classical orthogonal p olynomials we nd the TTRR which satisfy the p erturb ed ones. In Section 6, from the relation of the p erturb ed p olynomials P A n ( x ) as a linear combination of the classical ones, we nd the tridiagonal matrices asso ciated with the p erturb ed monic orthogonal p olinomial sequence (PMOPS) f P A n ( x ) g 1 n =0 as a rankone p erturbation of the tridiagonal matrices asso ciated with the classical monic orthogonal p olynomial sequence (CMOPS) f P n ( x ) g 1 n =0 . Finally, in Section 7 we nd the asso ciated p olynomials P (1) ;A n ( x ) corresp onding to f P A n ( x ) g 1 n =0 in terms of the asso ciated p olynomials P (1) n ( x ) corresp onding to f P n ( x ) g 1 n =0 and the classical ones. x 2 Some Preliminar Results. Firstly, we enclose some formulas for the classical Meixner, Kravchuk and Charlier p olynomials which are useful in order to obtain the generalized p olynomials orthogonal with resp ect to the linear functional U dened as a mo dication of the rst ones troughtout the addition of a mass p oint. All the formulas and other prop erties for the classical Meixner, Kravchuk and Charlier p olynomials can b e found in a lot of b o oks ( see for instance the excellent monograph Orthogonal Polynomials in Discrete Variables by A.F. Nikiforov, S. K. Suslov, V. B. Uvarov [11], Chapter 2.) We will use monic p olynomials, i.e., p olynomials with leading co ecient equal to 1. The classical orthogonal p olynomials of a discrete variable in the uniform lattice x ( s ) = s are the 2 p olynomial solution of a second order linear dierence equation of hyp ergeometric typ e  ( x ) 4 5 P n ( x ) +  ( x ) 4 P n ( x ) +  n P n ( x ) = 0 ; (1) where 5 f ( x ) = f ( x )  f ( x  1) ; 4 f ( x ) = f ( x + 1)  f ( x ) : Here  ( x ) and  ( x ) are p olynomials in x of degree at most 1 and 2, resp ectively, and  n is a constant. These p olynomials are orthogonal with resp ect to the linear functional L on the linear space of p olynomials with real co ecients dened as < L ; P > = X x 2 IN  ( x ) P ( x ) ; IN = f 0 ; 1 ; 2 ; ::: g ; (2) where  ( x ) is some non-negative function ( weight function ) supp orted in a countable set of the real line and such that 4 [  ( x )  ( x )] =  ( x )  ( x ) : The orthogonality relation is X x 2 IN P n ( x ) P m ( x )  ( x ) =  nm d 2 n ; (3) where d 2 n denotes the square of the norm of these classical p olynomials. The p olynomial solutions of equation (1) are uniquely determined, up to a normalized factor ( R n ), by the dierence analog of the Ro drigues formula (see [11] page 24 Eq.(2.2.7)): P n ( x ) = R n  ( x ) 5 n "  ( x + n ) n Y k =1  ( x + k ) # : (4) They satisfy a three term recurrence relation of the form xP n ( x ) = P n +1 ( x ) +  n P n ( x ) +  n P n  1 ( x ) ; n  0 P  1 ( x ) = 0 and P 0 ( x ) = 1 : (5) and the Christoel-Darb oux formula n  1 X m =0 P m ( x ) P m ( y ) d 2 m = 1 x  y P n ( x ) P n  1 ( y )  P n ( y ) P n  1 ( x ) d 2 n  1 n = 1 ; 2 ; 3 ; ::: : (6) We will consider the following three classical monic orthogonal p olynomials (CMOP) which are solutions of the dierence equation (1). I. The Meixner p olynomials, orthogonal with resp ect to the weight function  ( x ) supp orted on [0 ; 1 ), with  ( x ) = x;  ( x ) =    x (1   ) ; 0 <  < 1 ;  > 0 ;  n = n (1   ) ; and R n = 1 (   1) n ;  ( x ) =  x (  + x ) (  )(1 + x ) ; d 2 n = n !(  ) n  n (1   )  +2 n : I I. The Kravchuk p olynomials, orthogonal with resp ect to the weight function  ( x ) supp orted on [0 ; N ], with n  N  ( x ) = x;  ( x ) = N p  x 1  p ; 0 < p < 1 ;  n = n 1  p ; 3 and R n = ( p  1) n ;  ( x ) = p x N !(1  p ) N  x ( N + 1  x )(1 + x ) ; d 2 n = n ! N ! p n (1  p ) n ( N  n )! : I I I. The Charlier p olynomials, orthogonal with resp ect to the weight function  ( x ) supp orted on [0 ; 1 ), with  ( x ) = x; ;  ( x ) =   x ;  > 0 ;  n = n; and R n = (  1) n ;  ( x ) =  x e   (1 + x ) ; d 2 n = n !  n : They satisfy the so called structure relations x n 5 M  ; n ( x ) =  (1    n )   1 M  ; n  1 ( x ) + M  ; n ( x ) ; (7) x n 5 K p n ( x ) = p ( N  n + 1) K p n  1 ( x ) + K p n ( x ) ; (8) x n 5 C  n ( x ) = C  n  1 ( x ) + C  n ( x ) : (9) These classical p olynomials can b e represented as hyp ergeometric functions (see [11] page 49, section 2.7 ) M  ; n ( x ) = (  ) n  n (   1) n 2 F 1   n;  x      1  1   ; (10) K p n ( x ) = (  p ) n N ! ( N  n )! 2 F 1   n;  x  N     1 p  ; (11) C  n ( x ) = (   ) n 2 F 0   n;  x       1   ; (12) where the hyp ergeometric function is dened by p F q  a 1 ;a 2 ;:::;a p b 1 ;b 2 ;:::;b q j x ) = 1 X k =0 ( a 1 ) k ( a 2 ) k    ( a p ) k ( b 1 ) k ( b 2 ) k  ( b q ) k x k k ! ; ( a ) 0 := 1 ; ( a ) k := a ( a + 1)( a + 2)  ( a + k  1) ; k = 1 ; 2 ; 3 ; ::: As a consequence of these representations we can deduce M  ; n (0) =  n (   1) n ( n +  ) (  ) ; K p n (0) = (  p ) n N ! ( N  n )! ; C  n (0) = (   ) n : (13) We have proved (see [2] formula (26)) the following prop erty for the kernels of the Meixner p olynomials. K er M n  1 ( x; 0)  n  1 X m =0 M  ; m ( x ) M  ; m (0) d 2 m = (  1) n  1 (1   ) n +   1 n ! 5 M  ; n ( x ) : (14) It is straightforward to show that the kernels of K p n ( x ) and C  n ( x ) verify the following relations: K er K n  1 ( x; 0)  n  1 X m =0 K p m ( x ) K p m (0) d 2 m = ( p  1) 1  n n ! 5 K p n ( x ) ; (15) K er C n  1 ( x; 0)  n  1 X m =0 C  m ( x ) C  m (0) d 2 m = (  1) n  1 n ! 5 C  n ( x ) : (16) 4 x 3 The denition, orthogonal relation and representation as hyp ergeometric series. Consider the linear functional U on the linear space of p olynomials with real co ecients dened as < U ; P > = < L ; P > + AP (0) ; x 2 IN ; A  0 ; (17) where L is a classical moment functional (2) asso ciated to some classical p olynomials of a discrete variable. We will determine the monic p olynomials P A n ( x ) which are orthogonal with resp ect to the functional U and prove that they exist for all p ositive A (see (22) from b elow). To obtain this, we can write the Fourier expansion of such generalized p olynomials P A n ( x ) = P n ( x ) + n  1 X k =0 a n;k P k ( x ) ; (18) where P n denotes the classical monic orthogonal p olynomial (CMOP) of degree n . In order to nd the unknown co ecients a n;k we will use the orthogonality of the p olynomials P A n ( x ) with resp ect to U , i.e., < U ; P A n ( x ) P k ( x ) > = 0 8 k < n: Now putting (18) in (17) we nd: < U ; P A n ( x ) P k ( x ) > = < L ; P A n ( x ) P k ( x ) > + AP A n (0) P k (0) : (19) If we use the decomp osition (18) and taking into account the orthogonality of the classical orthogonal p olynomials with resp ect to the linear functional L , then the co ecients a n;k are given by: a n;k =  A P A n (0) P k (0) d 2 k : (20) Finally the equation (18) provides us the expression P A n ( x ) = P n ( x )  AP A n (0) n  1 X k =0 P k (0) P k ( x ) d 2 k : (21) From (21) we can conclude that the representation of P A n ( x ) exists for any p ositive value of the mass A . To obtain this it is enough to evaluate (21) in x = 0, 1 + A n  1 X k =0 ( P k (0)) 2 d 2 k ! P A n (0) = P n (0) 6 = 0 ; (22) and use the fact that 1 + A n  1 X k =0 ( P k (0)) 2 d 2 k > 0 n = 1 ; 2 ; 3 ; ::: From (22) we can deduce the values of P A n (0) as follows 5 P A n (0) = P n (0) 1 + A n  1 X k =0 ( P k (0)) 2 d 2 k : (23) Now in order to obtain an explicit expression for these p olynomials we need some properties of the kernels of the CMOP. In [2] we solved this problem for the classical Meixner p olynomials. In this work we will prove a similar result for Charlier and Kravchuk p olynomials. Doing some algebraic calculations in (21) and taking into account formulas (14)-(16) we obtain the following three expressions for the generalized p olynomials : For Meixner p olynomials: M  ;;A n ( x ) = M  ; n ( x )  AM  ;;A n (0) (  1) n  1 (1   ) n +   1 n ! 5 M  ; n ( x ) : (24) For Kravchuk p olynomials: K p;A n ( x ) = K p n ( x )  AK p;A n (0) ( p  1) n  1 n ! 5 K p n ( x ) : (25) For Charlier p olynomials: C ;A n ( x ) = C  n ( x )  AC ;A n (0) (  1) n  1 n ! 5 C  n ( x ) : (26) In the ab ove formula the values of the p olynomials in x = 0 could b e deduced from (23). Then, we obtain the following analytic expression for the Perturb ed Monic Orthogonal Polynomials (PMOP) P A n ( x ) for x 6 = 0 (when x = 0 we can use (23) ) M  ;;A n ( x ) = M  ; n ( x ) + B n 5 M  ; n ( x ) = ( I + B n 5 ) M  ; n ( x ) ; (27) K p;A n ( x ) = K p n ( x ) + A n 5 K p n ( x ) = ( I + A n 5 ) K p n ( x ) ; (28) C ;A n ( x ) = C  n ( x ) + D n 5 C  n ( x ) = ( I + D n 5 ) C  n ( x ) ; (29) where A n ; B n and D n are constants given by: B n = A  n (1   )   1 (  ) n n !(1 + AK er M n  1 (0 ; 0)) ; A n = A N ! n !( N  n )! p n (1  p ) 1  n (1 + AK er K n  1 (0 ; 0)) ; D n = A  n n !(1 + AK er C n  1 (0 ; 0)) : Remark : Using the Ro drigues formula (4), some extension of it follows in a straightforward way. M  ;;A n ( x ) = ( I + B n 5 )   n (1   ) n ( x + 1)  x (  + x ) 5 ( n )  x ( x +  + n ) ( x + 1)  ; K p;A n ( x ) = ( I + A n 5 )  (  1) n ( x + 1)( N  x + 1) p x (1  p )  x 5 ( n ) p x (1  p )  x ( x + 1)( N  x  n + 1)  ; 6 C ;A n ( x ) = ( I + D n 5 )  (  1) n ( x + 1)  x 5 ( n )  x + n ( x + 1)  : Now we can establish the following representation as hyp ergeometric functions for the generalized p olynomials: Prop osition 1 The orthogonal polynomials M  ;;A n ( x ) , K p;A n ( x ) and C ;A n ( x ) are, up to a constant factor, generalized hypergeometric functions. More precisely M  ;;A n ( x ) = (  ) n  n (   1) n 3 F 2   n;  x; 1+ xB  1 n  ; xB  1 n     1  1   ; (30) K p;A n ( x ) = n !(  p ) n N ! n !( N  n )! 3 F 2   n;  x; 1+ xA  1 n  N ; xA  1 n     1 p  ; (31) C ;A n ( x ) = (   ) n 3 F 1   n;  x; 1+ xD  1 n xD  1 n      1   : (32) Sketch of the Pro of: The pro of of this Prop osition is similar to the pro of for the Meixner case (see [2]). To obtain the desired result we need to put the hyp ergeometric representation of these p olynomials in formulas (27), (28) and (29) and do some algebraic calculations. Here the co ecients xA  1 n , xB  1 n and xC  1 n are real numb ers. In the case when they are nonp ositive integers we need to take the analytic continuation of the hyp ergeometric series (30), (31) and (32). It is straightforward to show that for A = 0 the hyp ergeometric functions (30), (31) and (32) yield to classical p olynomials (10), (11) and (12). x 4 A second order dierence equation. In [2] we proved that the Meixner p olynomials satisfy a second order dierence equation. To prove this result we only used that in the dierence equation of hyp ergeometric typ e for Meixner p olynomials the function  ( x ) is equal to x. Taking into account that, for Charlier and Kravchuk p olynomials,  ( x ) = x , then the following Theorem holds: Theorem 1 The polynomials M  ;;A n ( x ) ; K p;A n ( x ) and C ;A n ( x ) satisfy a second order linear dierence equation [ x + C ( C  n +  n   ( x ))]( x  1) 4 5 P A n ( x ) + ( x  1)  ( x ) 4 P A n ( x )+ + C [(  ( x )  C  n )(  n  1   ( x  1)) +  n (  n + C ) + ( x + C  n ) 4  ( x )] 4 P A n ( x ) +( x  1)  n P A n ( x ) + C  n [  n  1   ( x  1) + C ( 4  ( x ) +  n )] P A n ( x ) = 0 ; (33) where x = 0 ; 1 ; 2 ; :::; 5 f ( x ) = f ( x )  f ( x  1) ; 4 f ( x ) = f ( x + 1)  f ( x ) ; and by P A n we denote the generalized Meixner, Kravchuk or Charlier polynomials and C is the constant B n ; A n or D n , respectively, which is a function of n (see Section 3 ). The pro of of this Theorem for the Meixner case was given in [2]. Here we provide the pro of for b oth three cases. 7 Pro of: We will start from the representations (27), (28) and (29) for the generalized p olynomials P A n ( x ) = P n ( x ) + C 5 P n ( x ). Multiplying this expression by x , using the second order dierence equation that these classical p olynomials satisfy x 4 5 P n ( x ) +  ( x ) 4 P n ( x ) +  n P n ( x ) = 0 ; (34) and using the identity 5 P n ( x ) = 4 P n ( x ) 45 P n ( x ) we obtain xP A n ( x ) = ( x + C  n ) P n ( x ) + C ( x +  ( x )) 4 P n ( x ) : (35) Now if we apply the op erator 4 to (35), from (34) the equation x 4 P A n ( x ) = [ x  C  ( x )] 4 P n ( x )  C  n P n ( x ) (36) follows. In the same way if we apply in (36) the op erator 5 and using (34) we nd x ( x  1) 4 5 P A n ( x ) = =  [( x  1)  ( x ) + C  ( x )(  n   ( x  1)  1) + C x (  n + 4  ( x ))] 4 P n ( x )   [ x  1 + C (  n   ( x  1)  1)]  n P n ( x ) : (37) Now from (35),(36) and (37) the following determinant vanishes       xP A n ( x ) a ( x ) b ( x ) x 4 P A n ( x ) c ( x ) d ( x ) x ( x  1) 4 5 P A n ( x ) e ( x ) f ( x )       = 0 ; (38) where a ( x ) = ( x + C  n ) ; b ( x ) = C ( x +  ( x )) ; c ( x ) =  C  n ; d ( x ) = x  C  ( x ) ; e ( x ) =  [ x  1 + C (  n  1   ( x  1))]  n ; f ( x ) =  [( x  1)  ( x ) + C [  ( x )(  n  1   ( x  1)) + x (  n + 4  ( x ))] : Expanding the determinant in (38) by the rst column and dividing by x 2 ; the Theorem follows. The dierence equation of the previous theorem (33) takes the form: Meixner case M  ;;A n ( x ) f x + B n [(1   )( x + n + nB n )    ] g ( x  1) 4 5 M  ;;A n ( x )+ ( x  1)[    x (1   )] 4 M  ;;A n ( x )+ + B n f (1   )[   ( n + nB n + 2 x  1) + (1   )( x + n 2  ( x + nB n )( x + n ))+ +2 nB n ]    (1 +   ) g 4 M  ;;A n ( x ) + ( x  1) n (1   ) M  ;;A n ( x )+ nB n (1   )[(1   )( x + n + nB n  B n  1)  1    ] M  ;;A n ( x ) = 0 : (39) 8 Kravchuk case K p;A n ( x ) f x + A n [ x + n + nA n 1  p  N p 1  p ] g ( x  1) 4 5 K p;A n ( x )+ ( x  1)[ N p  x 1  p ] 4 K p;A n ( x )+ + A n f 1 1  p [ N p 1  p ( n + nA n + 2 x  1) + x + n 2  ( x + nA n )( x + n ) 1  p + +2 nA n ]  N p 1  p (1 + N p 1  p ) g 4 K p;A n ( x ) + n 1  p ( x  1) K p;A n ( x )+ nA n 1  p [ x + n + nA n  A n  1  N p 1  p  1] K p;A n ( x ) = 0 : (40) Charlier case C ;A n ( x ) [ x + D n ( x + n + nD n   )]( x  1) 4 5 C ;A n ( x )+ ( x  1)(   x ) 4 C ;A n ( x )+ + D n [ n 2 + x  2  + 2 nD n  ( n  x   )( x   + nD n )] 4 C ;A n ( x )+ +( x  1) nC ;A n ( x ) + nD n ( x + n    2 + nD n  D n ) C ;A n ( x ) = 0 : (41) x 5 Three Term Recurrence Relations. The generalized p olynomials satisfy a three term recurrence relation (TTRR) of the form xP A n ( x ) = P A n +1 ( x ) +  A n P A n ( x ) +  A n P A n  1 ( x ) ; n  0 P A  1 ( x ) = 0 and P A 0 ( x ) = 1 : (42) This is a simple consequence of their orthogonality with resp ect to a p ositive functional (see [6] or [11]). To obtain the explicit formula for the recurrence co ecients we can compare the co ecients of x n in the two sides of (42). Let b A n b e the co ecient of x n  1 in the expansion P A n ( x ) = x n + b A n x n  1 + ::: , then :  A n = b A n  b A n +1 . To calculate  A n is sucient to evaluate (42) in x = 0 and remark that P A n (0) 6 = 0. In order to obtain a general expression for the co ecient  A n we can use the formulas (27), (28) and (29) for the generalized p olynomials P A n ( x ) = P n ( x ) + C 5 P n ( x ), where C = B n ; A n or D n resp ectively. Doing some algebraic calculations we nd that b A n = b n + nC , where b n denotes the co ecient of the n  1 p ower in the classical p olynomials P n ( x ) = x n + b n x n  1 + ::: . Using these formulas and the main data [11] for classical p olynomials we obtain for generalized Meixner, Kravchuk and Charlier p olynomials the following TTRR co ecients : I Meixner p olynomials: b n = n   1 (  + n  1 2  + 1  ) ;  n = n +  ( n +  ) 1   ; b A n = n   1 (  + n  1 2  + 1  ) + A  n (1   )   1 (  ) n ( n  1)!(1 + AK er M n  1 (0 ; 0)) ; 9