On the linearization problem involving Pochhammer symbols and their q-analogues
Abstract
In this paper we present a simple recurrent algorithm for solving the linearization problem involving some families of q-polynomials in the exponential lattice x(s)=c1qs+c3. Some simple examples are worked out in detail.
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ON THE LINEARIZATION PROBLEM INVOLVING POCHHAMMER SYMBOLS AND THEIR qANALOGUES. 1 R. Alvarez-No darse a;b , N. R. Quintero , and A. Ronveaux d a Departamento de Analisis Matematio. Universidad de Sevil la. Apdo. 1160, E-41080, Sevil la, Spain b Instituto Carlos I de Fsia Teoria y Computaional. Universidad de Granada. E-18071 Granada, Spain Departamento de Matematias. Universidad Carlos III de Madrid. Ave. Universidad 30, 28911, Leganes, Madrid, Spain d Mathematial Physis, Faultes Universitaires Notre-Dame de la Paix, B-5000 Namur, Belgium. Key words and phrases: Po hhammer and qPo hhammer symb ols, linearization problem. AMS (MOS, 1991) sub jet lassiation: 33C45, 33D45 July 21, 1998. Revised February 1, 1999 Abstrat In the this pap er we present a simple reurrent algorithm for solving the Linearization Problem involving some families of qp olynomials in the exp onential lattie x ( s ) = 1 q s + 3 . Some simple examples are worked out in details. 1 Intro dution. Given three families of p olynomials, denoted by P n ( x ), Q m ( x ) and R j ( x ), of degree exatly equal to resp etively n , m and j , the Linearization Problem asks to ompute the so-alled linearization oeÆients L mj n dened by the relation: Q m ( x ) R j ( x ) = m + j X n =0 L mj n P n ( x ) : The study of suh a problem has known an inreasing interest in the last few years. Sp eial emphasis was given to the lassial ontinuous (Hermite, Laguerre, Jaobi and Bessel) [8, 9, 11, 15, 19, 20 , 21, 24, 26 , 27, 28℄ and the disrete ases (Charlier, Meixner, Kravhuk and Hahn) [7, 9, 12 , 14, 27 ℄. The main aim of the present pap er is to show that the ideas given in [14, 26℄ an b e extended in a very easy way to the qp olynomials on the exp onential lattie x ( s ) = 1 q s + 3 . If fat, if P n ( x ), Q m ( x ) and R j ( x ) are p olynomials in x ( s ) = 1 q s + 3 , then it is p ossible to nd a reurrene relation for the linearization o eÆients L mj n , whih is an alternative approah to the one given in [5℄. This approah requires the knowledge of the so-alled struture relation and three-term reurrene relation 1 E-mails: rania.es, kintermath.u3m.es, Andre.Ronveauxfundp.a.b e 1
for the p olynomials P n ( x ) as well as the seond order dierene equation whih satisfy the other two p olynomials Q m ( x ) and R j ( x ). In this pap er we will present an algorithm for omputing the ab ove o eÆients, showing, as an example, the ase when the p olynomials P n ( x ), Q m ( x ) and R j ( x ) are the three Po hhammer and qPo hhammer symb ols. In b oth ases the solutions are given expliitly. The obtained expression an b e used for solving more \ompliated" examples. The struture of the pap er is as follows. In Setion 2, a general algorithm for nding a reurrene relation for the L mj n , is presented. In Setion 3, two sp eial ases, of partiular imp ortane, are worked out, namely, the linearization of a pro dut of two qPo hhammer or qStirling p olynomials in terms of single qPo hhammer or qStirling p olynomials, resp etively. Finally, some more ompliated examples involving the Charlier p olynomials and their qanalogues in the lattie x ( s ) = q s are presented. 2 General algorithm for solving the linearization problem. In this setion we will present a general algorithm to nd a reurrene relation for the linearization o eÆients L mj n in the expansion Q m ( x ( s )) R j ( x ( s )) = m + j X n =0 L mj n P n ( x ( s )) ; x ( s ) = 1 q s + 3 ; (2.1) where 1 , 3 and q are onstants, Q m ( x ( s )) Q m ( s ) q and R j ( x ( s )) R j ( s ) q are p olynomials whih satisfy a seond order dierene equation of the form a ( s ) Q m ( s + 1) q + b ( s ) Q m ( s ) q + ( s ) Q m ( s 1) q = 0 ; (2.2) and ( s ) R j ( s + 1) q + ( s ) R j ( s ) q + ( s ) R j ( s 1) q = 0 ; (2.3) resp etively. A sp eial ase of suh p olynomials are the qp olynomials of hyp ergeometri typ e [3, 22 , 23℄, whih satisfy the dierene equation ( s ) 4 4 x ( s 1 2 ) 5 y ( s ) 5 x ( s ) + ( s ) 4 y ( s ) 4 x ( s ) + y ( s ) = 0 ; 5 f ( s ) = f ( s ) f ( s 1) ; 4 f ( s ) = f ( s + 1) f ( s ) : (2.4) Obviously, the Eq. (2.4) is of the typ e (2.2) ( y Q m ), with a ( s ) = ( s ) + ( s ) 4 x ( s 1 2 ) ; ( s ) = ( s ) 4 ( s ) 5 x ( s ) ; b ( s ) = 4 x ( s 1 2 ) 4 x ( s ) a ( s ) ( s ) : (2.5) Notie that when x ( s ) = 1 q s + 3 , 5 x ( s ), 4 x ( s ) and 4 x ( s 1 2 ) are p olynomials of rst degree in x ( s ), then, in this ase, the funtions a , b , , , and in (2.2)-(2.3) are p olynomials of 2th degree at most. 2
In the following, we will use the op erators T and I dened as follows T : IP ! IP T p ( s ) = p ( s + 1) ; I : IP ! IP I p ( s ) = p ( s ) : Using the ab ove op erators, we an rewrite the Eqs. (2.2)-(2.3) in the form a ( s + 1) T 2 Q m ( s ) q + b ( s + 1) T Q m ( s ) q + ( s + 1) I Q m ( s ) q = 0 ; (2.6) and ( s + 1) T 2 R j ( s ) q + ( s + 1) T R j ( s ) q + ( s + 1) I R j ( s ) q = 0 : (2.7) Other imp ortant examples of qp olynomials are the so-alled qPo hhammer symb ols and the qStirling p olynomials (see setion 3 of the present work for more details) whih satisfy a rst order dierene equation of typ e (2.6)- (2.7). 2.1 The fourth order dierene equation for the p olynomials Q m ( s ) q R j ( s ) q . Using the two equations (2.6) and (2.7), it is easy to prove that the p olynomials u ( s ) q Q m ( s ) q R j ( s ) q , satisfy a fourth order dierene equation of the form L 4 u ( s ) p 4 ( s ) T 4 u ( s ) q + p 3 ( s ) T 3 u ( s ) q + p 2 ( s ) T 2 u ( s ) q + p 1 ( s ) T u ( s ) q + p 0 ( s ) I u ( s ) q : (2.8) To prove this, we will follow the works [13, 14℄. The idea is the following. Sine (2.6)-(2.7), a ( s + 1) ( s + 1) T 2 u ( s ) = = [ b ( s + 1) T Q m ( s ) q + ( s + 1) I Q m ( s ) q ℄ [ ( s + 1) T R j ( s ) q + ( s + 1) I R j ( s ) q ℄ ; whih an b e rewritten as L 2 u ( s ) a ( s + 1) ( s + 1) T 2 u ( s ) b ( s + 1) ( s + 1) T u ( s ) ( s + 1) ( s + 1) I u ( s ) = = b ( s + 1) ( s + 1) [ T Q m ( s ) q I R j ( s ) q ℄ + ( s + 1) ( s + 1) [ I Q m ( s ) q T R j ( s ) q ℄ = = l 1 ( s ) [ T Q m ( s ) q I R j ( s ) q ℄ + l 2 ( s ) [ I Q m ( s ) q T R j ( s ) q ℄ : Next, we hange in the last expression s ! s + 1, and substitute in the right-hand side the expression T 2 Q m ( s ) q and T 2 R j ( s ) q , using the Eqs. (2.6)-(2.7), resp etively. This allows us to rewrite the resulting expression in the form M 3 u ( s ) = m 1 ( s ) [ T Q m ( s ) q I R j ( s ) q ℄ + m 2 ( s ) [ I Q m ( s ) q T R j ( s ) q ℄ ; where M 3 is a dierene op erator of third order (there is one term prop ortional to T 3 ), m 1 and m 2 are known funtions of s . Rep eating the same pro edure, but now starting from the ab ove equation we obtain N 4 u ( s ) = n 1 ( s ) [ T Q m ( s ) q I R j ( s ) q ℄ + n 2 ( s ) [ I Q m ( s ) q T R j ( s ) q ℄ : Then L 2 u ( s ) l 1 ( s ) l 2 ( s ) M 3 u ( s ) m 1 ( s ) m 2 ( s ) N 4 u ( s ) n 1 ( s ) n 2 ( s ) = 0 : (2.9) Expanding the determinant from the rst olumn, the Eq. (2.8) holds. Remark: The ab ove equation (2.9), and its pro of, remains true for any lattie funtion x ( s ) and not only for the exp onential lattie x ( s ) = 1 q s + 3 . 3
2.2 The generalized linearization algorithm As b efore, we will supp ose that Q m ( s ) q and R j ( s ) q satisfy the equations (2.6) and (2.7), resp etively, and that P n ( s ) q satisfy the so-alled struture relation in the exp onential lattie x ( s ) = 1 q s + 3 and a three-term reurrene relation. An example of the latest are, again, the qhyp ergeometri orthogonal p olynomials in the aforementioned lattie whih satisfy the struture relation [4℄ (for the ase x ( s ) = q s see [6℄) [ ( s ) + ( s ) 4 x ( s 1 2 )℄ 4 P n ( s ) q 4 x ( s ) = S n P n +1 ( s ) q + T n P n ( s ) q + R n P n 1 ( s ) q ; (2.10) or, written in its equivalent form ( s ) T P n ( s ) q = n +2 X k = n 2 A k ( n ) P k ( s ) q ; ( s ) = ( s ) + ( s ) 4 x ( s 1 2 ) ; (2.11) as well as the three-term reurrene relation (TTRR) x ( s ) P n ( s ) q = n P n +1 ( s ) q + n P n ( s ) q + n P n 1 ( s ) q ; P 1 ( x ) 0 ; n 0 : (2.12) To obtain (2.11) from (2.10) we use the fats that 4 x ( s ) is a p olynomial of rst degree in x ( s ) (whih is not valid in general for any lattie x ( s )), ( s ) + ( s ) 4 x ( s 1 2 ) is a p olynomial of degree two in x ( s ), and the TTRR (2.12). From the ab ove expression (2.11), one easily obtains that ( s )( s + 1) T 2 P n ( s ) q = n +4 X k = n 4 ~ A k ( n ) P k ( s ) q ; ( s )( s + 1)( s + 2) T 3 P n ( s ) q = n +6 X k = n 6 ^ A k ( n ) P k ( s ) q ; ( s )( s + 1)( s + 2)( s + 3) T 4 P n ( s ) q = n +8 X k = n 8 A k ( n ) P k ( s ) q : (2.13) To obtain a reurrene relation for the linearization o eÆients we an do the following: Sine (2.8), L 4 Q m ( s ) q R j ( s ) q = 0, then applying L 4 to b oth sides of (2.1), we nd 0 = m + j X n =0 L mj n ( s )( s + 1)( s + 2)( s + 3) L 4 P n ( x ( s )) : Taking into aount that L 4 is a fourth degree op erator with p olynomial o eÆients, and 4
using the struture relation (2.11) as well as (2.13) we nd 0 = m + j X n =0 L mj n ( p 4 ( s ) n +8 X k = n 8 A k ( n ) P k ( s ) q + p 3 ( s )( s + 3) n +6 X k = n 6 ^ A k ( n ) P k ( s ) q + + p 2 ( s )( s + 2)( s + 3) n +4 X k = n 4 ~ A k ( n ) P k ( s ) q + + p 1 ( s )( s + 1)( s + 2)( s + 3) n +2 X k = n 2 A k ( n ) P k ( s ) q + +( s )( s + 1)( s + 2)( s + 3) p 0 ( s ) P n ( s ) q ) ; from where, and by taking into aount that ( s + k ), k = 0 ; 1 ; 2 ; 3, is a p olynomial of degree two in x ( s ) = 1 q s + 3 , as well as the TTRR (2.12) we obtain that the o eÆients L mj n satisfy a reurrene relation of the form r X k =0 k ( i; j; n ) L mj n + k = 0 : (2.14) In general, the present algorithm may not give the minimal order reurrene for the linearization o eÆients. To get the order r minimal it is neessary to use more sp ei prop erties of the families of p olynomials involved in (2.1). Remark: Notie that the present algorithm also works in the ase when the pro dut Q m ( x ( s )) R j ( x ( s )) satisfy a k th-linear dierene equation with p olynomial o eÆients (not neessary of order 4 as in (2.9)), so it an b e used for solving more general linearization problems, e. g., linearization problems involving the pro dut of three or more q p olynomials. 3 Examples In this setion we will work out some examples. For simpliity we will onsider the ase when the involved p olynomials Q m and R j satisfy rst order dierene equations, i.e., equation of the form (2.6) and (2.7) with a ( s ) = ( s ) 0. 3.1 Linearization of a pro dut of two qPo hhammer symb ols. Let us dene the quantities ( s ) q by ( s ) q = q s 1 q 1 = q s 2 1 [ s ℄ q ; (3.1) and let [( s ) q ℄ n , the qPo hhammer symb ol, b e dened by [( s ) q ℄ n = ( s ) q ( s + 1) q ( s + n 1) q = n 1 Y k =0 q s + k 1 q 1 : (3.2) 5
Notie that [( s ) q ℄ n is a p olynomial of degree exatly equal n in q s . The p olynomials [( s ) q ℄ n satisfy the following dierene equation ( s ) q [( s + 1) q ℄ n ( s + n ) q [( s ) q ℄ n = 0 ; (3.3) and a reurrene relation ( s ) q [( s ) q ℄ n q n [( s ) q ℄ n +1 + q n ( n ) q [( s ) q ℄ n = 0 : (3.4) Notie also that [( s ) q ℄ n = ( q s ; q ) n (1 q ) n ; where ( a ; q ) n = n 1 Y k =0 (1 a q k ) : (3.5) Obviously, in the limit q ! 1, the qPo hhammer symb ol [( s ) q ℄ n , b eomes into the lassial Po hhammer symb ol ( s ) n = s ( s + 1) ( s + n 1). Sine the pro dut [( s ) q ℄ i [( s ) q ℄ j is a p olynomial in q s , it an b e represented as a linear ombination of the single qPo hhammer symb ols [( s q )℄ n . In partiular, [( s ) q ℄ i [( s ) q ℄ j = i + j X n =0 L ij n ( q )[( s q )℄ n : (3.6) In order to to obtain the reurrene relation for the linearization o eÆients L ij n in (3.6) we apply the op erator ( s ) 2 q T ( s + i ) q ( s + j ) q I (3.7) to b oth sides of (3.6). Using formula (3.3) we obtain the following expression 0 = i + j X n =0 L ij n " q s 1 q 1 2 T [( s q )℄ n q s + i 1 q 1 ! q s + j 1 q 1 ! [( s q )℄ n # : (3.8) Taking into aount the Eq. (3.3) for the qPo hhammer symb ol, we nd 0 = i + j X n =0 L ij n [( s q )℄ n " q s 1 q 1 q s + n 1 q 1 ! q s + i 1 q 1 ! q s + j 1 q 1 !# = = i + j X n =0 L ij n [( s q )℄ n [( s ) q ( s + n ) q ( s + i ) q ( s + j ) q ℄ : Next, we will rewrite the expression inside the quadrati brakets using the identity ( s + n ) q = q n ( s ) q + ( n ) q ; to obtain 0 = i + j X n =0 L ij n [( s q )℄ n n ( s ) 2 q [ q n q i + j ℄ + ( s ) q [( n ) q q i ( j ) q q j ( i ) q ℄ ( i ) q ( j ) q o ; 6
from where, using Eq. (3.4), we arrive to the expression i + j X n =0 L ij n n q 2 n 1 [ q n q i + j ℄[( s q )℄ n +2 + + ( n ) q q i ( j ) q q j ( i ) q q n ( q n q i + j ) q 2 n 1 ( n + 1) q + q 2 n ( n ) q [( s q )℄ n +1 + + h ( q n q i + j ) q 2 n ( n ) 2 q ( n ) q q i ( j ) q q j ( i ) q q n ( i ) q ( j ) q i [( s q )℄ n o = = i + j X n =0 L ij n n q 2 n 1 [ q n q i + j ℄[( s q )℄ n +2 q n 1 ( n + 1) q + q i + j 1 2 n 1 + q n + j +1 ( j ) q + q n + i +1 ( i ) q 2( n + 1) q [( s q )℄ n +1 q i + j 2 n ( n ) q q n + j ( j ) q ( n ) q q n + i ( i ) q [( s q )℄ n o = 0 : The ab ove equation leads us to the following three-term reurrene relation for the linearization o eÆients L ij n A n L ij n 2 + B n L ij n 1 + C n L ij n = 0 ; (3.9) where A n = q 2 n +3 [ q n 2 q i + j ℄ ; B n = q n ( n ) q q i + j +1 n q j ( j ) q + q i ( i ) q q n ( n ) q q n +1 ( n 1) q ; C n = q i + j q n ( n ) q q j ( j ) q q n ( n ) q q i ( i ) q ; (3.10) with the initial onditions L ij i + j +1 = 0 and L ij i + j = q ij . To solve the ab ove reurrene we apply the algorithm qHyp er [1 , 2, 25℄ whih allows us to nd an equivalent two-term reurrene relation for the linearization o eÆients. Namely, L ij n +1 = q k 1 ( i + j n ) q ( i n 1) q ( j n 1) q L ij n ; (3.11) so that, L ij n = ( 1) i + j n q i ( i +1)+ j ( j +1) n ( n +1) 2 [( j ) q ℄ i + j n [( i ) q ℄ i + j n ( i + j n ) q ! ; (3.12) for n max( i; j ) and vanishes otherwise. Notie that, in the limit q ! 1, the ab ove reurrene relations (3.9)-(3.11) transform into a two-term reurrene relations for the standard Po hhammer symb ols ( s ) n of the form ( k i j 1) L ij n 1 ( k 2 ( i + j ) k + ij ) L ij n = 0 ; L ij i + j +1 = 0 ; L ij i + j = 1 ; 7
whih solution L ij n = 8 > > > < > > > : ( 1) i + j + n ( j ) i + j n ( i ) i + j n ( i + j n )! n max( i; j ) 0 otherwise ; orresp onds to (3.12) in the limit q ! 1. 3.2 Linearization of a pro dut of two qStirling p olynomials. Let us dene the qStirling p olynomials or qfalling fatorials ( s ) [ n ℄ q , by ( s ) [ n ℄ q = ( s ) q ( s 1) q ( s n + 1) q = n 1 Y k =0 q s k 1 q 1 (3.13) Also we will use the notation ( s ) [ n ℄ q = ( q s ; q ) [ n ℄ (1 q ) n ; ( a ; q ) [ n ℄ = (1 a )(1 aq 1 ) (1 aq n +1 ) : (3.14) These quantities ( s q ) [ n ℄ are losely related to the q-Stirling numbers ~ S q ( n; k ) ; s q ( n; k ) [30℄ by formulas ( s ) n q = n X k =0 ~ S q ( n; k )( s q ) [ n ℄ ; ( s ) [ n ℄ q = n X k =0 s q ( n; k )( s ) k q ; (3.15) and satisfy the following dierene equation ( s ) q ( s 1) [ n ℄ q ( s n ) q ( s ) [ n ℄ q = 0 ; (3.16) as well as the reurrene relation ( s ) q ( s ) [ n ℄ q q n ( s ) [ n +1℄ q ( n ) q [( s ) q ℄ [ n ℄ = 0 : (3.17) Again, sine the pro dut ( s ) [ i ℄ q ( s ) [ j ℄ q is a p olynomial in x ( s ) = q s , it an b e represented as a linear ombination of the qfalling fatorials ( s q ) [ n ℄ . In partiular, ( s ) [ i ℄ q ( s ) [ j ℄ q = i + j X n =0 ~ L ij n ( q )( s q ) [ n ℄ : (3.18) To obtain the reurrene relation for the linearization o eÆients ~ L ij n in (3.18) we apply the op erator ( s ) 2 q T 1 ( s i ) q ( s j ) q I (3.19) to b oth sides of (3.18) and do similar alulations as b efore but now using the equations (3.16) and (3.17), resp etively. This leads us to the expression i + j X n =0 ~ L ij n n q 2 n +1 [ q n q i j ℄( s q ) [ n +2℄ + + ( n + 1) q + q n i j [( j ) q + ( i ) q ( n ) q ( n + 1) q ℄ ( s q ) [ n +1℄ q i j [( n ) q ( j ) q ℄ [( n ) q ( i ) q ℄ ( s q ) [ n ℄ o = 0 ; 8
whih allows us to obtain the following three-term reurrene relation for the linearization o eÆients ~ L ij n ~ A n ~ L ij n 2 + ~ B n ~ L ij n 1 + ~ C n ~ L ij n = 0 ; (3.20) where ~ A n = q 2 n 3 [ q n +2 q i j ℄ ; ~ B n = ( n ) q + q n i j 1 [( j ) q + ( i ) q ( n 1) q ( n ) q ℄ ; ~ C n = q i j [( n ) q ( j ) q ℄ [( n ) q ( i ) q ℄ ; (3.21) with the initial onditions ~ L ij i + j +1 = 0 and ~ L ij i + j = q ij . Again, applying to the ab ove reurrene relation the algorithm qHyp er [1 , 2, 25 ℄ we nd that the o eÆients ~ L ij n satisfy an equivalent two-term reurrene relation ~ L ij n +1 = q ( i + j n ) q ( i n 1) q ( j n 1) q ~ L ij n ; (3.22) so that, ~ L ij n = q i + j + ij n [( j ) q ℄ i + j n [( i ) q ℄ i + j n ( i + j n ) q ! ; for n max( i; j ) ; (3.23) and vanishes otherwise. We want to p oint out here that the ab ove reurrene relation (3.20), as well as its solution (3.23), an b e obtained from the previous equations (3.9) and (3.12). In order to do that we use the identity ( s ) [ n ℄ q = ( 1) n q n [( s ) q 1 ℄ n ; (3.24) and substitute it in (3.6). Comparing the obtained expression with (3.18) we nd ~ L ij n ( q ) = ( 1) i + j n q n i j L ij n ( q 1 ) : (3.25) The ab ove three-term reurrene relation (3.20) in the limit q ! 1, transforms into a two-term reurrene relation for the lassial Stirling p olynomials ( s ) [ n ℄ = s ( s 1) ( s n + 1), of the form ( j + i + 1 k ) ~ L ij n 1 ( k 2 ( i + j ) k + ij ) ~ L ij n = 0 ; L ij i + j +1 = 0 ; L ij i + j = 1 ; (3.26) whih have the solution ~ L ij n = 8 > > > < > > > : ( j ) i + j n ( i ) i + j n ( i + j n )! n max( i; j ) 0 otherwise ; (3.27) and that is in agreement with (3.23). 9