Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials
Abstract
Starting from the second-order difference hypergeometric equation satisfied by the set of discrete orthogonal polynomials ∗pn∗, we find the analytical expressions of the expansion coefficients of any polynomial rm(x) and of the product rm(x)qj(x) in series of the set ∗pn∗. These coefficients are given in terms of the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric polynomials. Here qj(x) denotes an arbitrary discrete hypergeometric polynomial of degree j. The particular cases in which ∗rm∗ corresponds to the non-orthogonal families ∗xm∗, the rising factorials or Pochhammer polynomials ∗(x)m∗ and the falling factorial or Stirling polynomials ∗x[m]∗ are considered in detail. The connection problem between discrete hypergeometric polynomials, which here corresponds to the product case with m = 0, is also studied and its complete solution for all the classical discrete orthogonal hypergeometric (CDOH) polynomials is given. Also, the inversion problems of CDOH polynomials associated to the three aforementioned nonorthogonal families are solved.
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Modified Clebsch-Gordan-type expansions for products of discrete hypergeometric polynomials. 1 Renato Alvarez-No darse Departamento de Matematicas, Universidad Carlos I I I de Madrid c/ Butarque 15, 28911, Leganes, Madrid, Spain and Instituto Carlos I de Fsica Teorica y Computacional, Universidad de Granada, 18071 Granada, Spain R. J. Ya~nez Departamento de Matematica Aplicada and Instituto Carlos I de Fsica Teorica y Computacional, Universidad de Granada, 18071 Granada, Spain and Jesus S. Dehesa Departamento de Fsica Mo derna and Instituto Carlos I de Fsica Teorica y Computacional, Universidad de Granada, 18071 Granada, Spain Decemb er 14, 1997 Abstract Starting from the second-order dierence hyp ergeometric equation satised by the set of discrete orthogonal p olynomials f p n g , we nd the analytical expressions of the expansion co ecients of any p olynomial r m ( x ) and of the pro duct r m ( x ) q j ( x ) in series of the set f p n g . These co ecients are given in terms of the p olynomial co ecients of the second-order dierence equations satised by the involved discrete hyp ergeometric p olynomials. Here q j ( x ) denotes an arbitrary discrete hyp ergeometric p olynomial of degree j . The particular cases in which f r m g corresp onds to the nonorthogonal families f x m g , the rising factorials or Po chhammer polynomials f ( x ) m g and the falling factorial or Stirling p olynomials f x [ m ] g are considered in detail. The connection problem b etween discrete hyp ergeometric p olynomials, which here corresp onds to the pro duct case with m = 0, is also studied and its complete solution for all the classical discrete orthogonal hyp ergeometric (CDOH) p olynomials is given. Also, the inversion problems of CDOH p olynomials associated to the three aforementioned non-orthogonal families are solved. Key words and phrases : discrete p olynomials, connection and linearization problems, discrete inversion formulas, second-order dierence equations. AMS (MOS, 1991) subject classification : 33D45, 33E30 PACS numbers: 02.30.Mv 1 E-Mails: [email protected]c3m.es, [email protected], [email protected]gr.es 1
1 Intro duction. The hyp ergeometric p olynomials of a discrete variable are ob jects which not only are imp ortant per se in the framework of the theory of sp ecial functions but also they play a very relevant role in numerous physical and mathematical elds which range from quantum mechanics, theory of group representations and probability theory to numerical analysis, the theory of vibrating strings, the study of random walks with discrete time pro cesses and the theory of Sturm-Liouville dierence equations as p ointed out by numerous authors [7, 21 ] and particularly the excellent monograph of A.F. Nikiforov, S.K. Suslov and V.B. Uvarov [41 ]. This is the case not only for the so-called classical discrete orthogonal p olynomials (Hahn, Meixner, Kravchuk and Charlier) but also for discrete orthogonal sets other than the classical ones; see e.g. [12 , 24, 39, 49 ]. The expansion of any arbitrary discrete p olynomial r m ( x ) in series of a general (alb eit xed) set of discrete hyp ergeometric p olynomial f p n ( x ) g is a matter of great interest, not yet solved save for some particular classical cases, as briey summarized by Askey [8] and Gasp er [17 , 18] up to the middle of seventies and by Ronveaux et al [45, 48], since then up to now. This is particulary true for the deep er problem of linearization of a pro duct of any two discrete p olynomials. Usually, the determination of the expansion co ecients in these particular cases required a deep knowledge of sp ecial functions and, at times, ingenious induction arguments based in the three-term recurrence relation of the involved orthogonal p olynomials [13, 17, 18 , 26, 51, 52 ]. Only recently, general and widely applicable strategies b egin to app ear [25 , 30, 37, 47 ]. Markett [37] for symmetric orthogonal p olynomials, has designed a metho d which b egins with the three-term recurrence relation of the involved orthogonal p olynomial system to set up a partial dierence equation for the (orthogonal) p olynomial, in case of connection problems, or for the pro duct of two (orthogonal) p olynomials, in case of linearization problems, to b e expanded; then, this equation has to b e solved in terms of the initial data. Ronveaux et al. [47, 48] for classical and semiclassical orthogonal p olynomials and Lewanowicz [30, 34] for classical orthogonal p olynomials have prop osed alternative, simpler techniques of the same typ e although they require the knowledge of not only the recurrence relation but also the dierential-dierence relation or/and the second-order dierence equation, resp ectively, satised by the p olynomials of the orthogonal set of the expansion problem in consideration. See also [5 , 11 , 23, 31 , 45, 46 ] for further description and applications of this metho d in the discrete case, and [19 , 32] in the continuous case as well as [3, 33] for the q -discrete orthogonality. Ko epf and Schmersau [25] has prop osed a computer-algebra-based metho d which, starting from the second order dierence hyp ergeometric equation, pro duces by symb olic means and in a recurrent way the expansion co ecients of the classical discrete orthogonal hyp ergeometric p olynomials (CDOHP) in terms of the falling factorial p olynomials (already obtained analytically by Lesky [28]; see also [41], [42 ]) as well as the expansion co ecients of its corresp onding inverse problem. The combination of these two simple expansion problems allows these authors to solve the connection problems within each sp ecic CDOHP set. All these four metho ds provide the expansion co ecients via recursion relationships, what is very useful for the symb olic and/or numerical computation of its values. However, in general, these relationships cannot b e analytically solved; so that, in practice, closed 2
expressions for the expansion co ecients are only obtained, at times, by symb olic means. The purp ose of this pap er is to describ e a general and constructive approach to solve the expansion formulas of the typ e r m ( x ) = m X n =0 c mn p n ( x ) ; (1.1) and r m ( x ) q j ( x ) = m + j X n =0 c j mn p n ( x ) ; (1.2) where r m ( x ) and q j ( x ) are any m th-degree and j th-degree discrete hyp ergeometric p olynomials, and f p n g denotes an arbitrary set of discrete orthogonal hyp ergeometric p olynomials. Expansions of typ e (1.1) are usually called as connection or pro jection formulas while those of typ e (1.2) are referred to generalized linearization formulas or mo died expansions of Clebsch-Gordan typ e [43]. Here the name Clebsch-Gordan is attached b ecause its structure is similar to the Clebsch-Gordan series for spherical functions [15 ]. Usually the Clebsch-Gordan expansion [15] or linearization relation involves p olynomials which b elongs to the same system (i.e., they are all three Hahn p olynomials, or all Meixner, etc). When this is not the case, we referred to mo died Clebsch-Gordan expansion or generalized linearization [43 ]. For the sake of completeness and notation, let us also mention that expansions (1.1) are called as inversion formulas when the p olynomials r m ( x ) b elongs to any of the following non-orthogonal families: the p ower p olynomials f x m g , the rising factorials or Po chhammer p olynomials f ( x ) m g and the falling factorials or Stirling p olynomials f x [ m ] g . The only prerequisite of our approach is the knowledge of the second order dierence equation satised by the involved hyp ergeometric p olynomials. The resulting expansion co ecients are given in a compact, analytic, closed and formally simple form in terms of the p olynomial co ecients of the corresp onding second-order dierence equation(s). Then, contrary to Market's, Ronveaux et al's and Lewanowicz's metho ds we do not require information ab out any kind of recurrence relation ab out the involved discrete hyp ergeometric p olynomials nor we need to solve any partial dierence equation for the p olynomial(s) to b e expanded, or \high" order recurrence relation for the connection co ecients themselves. Let us also underline that, opp osite to Ko epf and Schmersau's metho d, we do not use any symb olic means, as well as we directly provide the expansion co ecients in a single step. The structure of the pap er is the following. Firstly, in Section 2, we collect the basic background [41 ] used in the rest of the work; namely, the second-order hyp ergeometric dierence equation and its p olynomial solutions (called as discrete hyp ergeometric p olynomials) as well as the main data of the four classical sets of orthogonal discrete p olynomials (Hahn, Meixner, Kravchuk, and Charlier) in the monic form and the principal prop erties of the aforementioned non-orthogonal families f ( x ) m g and f x [ m ] g . Then, in Section 3, the co ecients of the expansion (1.1) are given explicitly in terms of the p olynomial co- ecients of the hyp ergeometric dierence equation satised by the orthogonal set f p n g . Also, as a consequence of the resulting expression, the inversion formulas asso ciated to the Po chhammer, Stirling and the p ower p olynomials are fully solved in the subsection 3.1 and then they are applied to the four classical discrete orthogonal sets. 3
In Section 4 the co ecients of the linearization formula (1.2) are found in a fully analytical way in terms of the p olynomial co ecients of the second-order dierence hyp ergeometric equations satised by the p olynomials p n ( x ) and q j ( x ). Notice that f q j g is not necessarily an orthogonal set, neither r m ( x ) is obliged to have a hyp ergeometric character, what widely extends the linearization formulas considered in the literature [5, 8, 9, 11 , 17 , 23 , 25, 30 , 31 , 37, 45 , 46 , 47, 48 ]. Indeed, most authors study linearizaton formulas b etween classical discrete p olynomials, usually within the same family (see e.g. [10, 11, 14 , 53 ] save some of them, who nd a few other formulas which either involve p olynomials of diferent classical families [17, 45 ] or include one of the aforementioned non-orthogonal families together with p olynomials of the same classical system [11]. The linearization formulas (1.2) corresp onding to the sp ecial cases in which r m ( x ) = ( x ) m , x [ m ] and x m , are given in subsection 4.1. In Section 5 the connection problem b etween discrete hyp ergeometric p olynomials is worked out in detail as the particular case m = 0 of the linearization formula fully solved in the previous section. The resulting expressions are used to explicitly obtain the connection formulas b etween p olynomials of each of the four classical families and b etween all of its p ossible pairs. This includes also the Hahn system, what we underline b ecause no general results of these typ e can b e encountered in the literature. Finally, some concluding remarks and a numb er of references are given. 2 \Discrete" preliminaries. Here we collect the basic background [40, 41 ] on hyp ergeometric discrete p olynomials and rising and falling factorials needed in the rest of the work. 2.1 The discrete hyp ergeometric p olynomials. Let us consider the second-order dierence equation of hyp ergeometric-typ e [40, 41], i.e., the equation ( x ) 5 4 y ( x ) + ( x ) 4 y ( x ) + y ( x ) = 0 ; (2.1) where ( x ) and ( x ) are p olynomials of degree not greater than 2 and 1, resp ectively, and is a constant. This equation can b e written in the self-adjoint form 4 [ ( x ) ( x ) 5 y ( x )] + ( x ) y ( x ) = 0 ; (2.2) where the function ( x ) satises the Pearson-typ e dierence equation 4 [ ( x ) ( x )] = ( x ) ( x ) : (2.3) The solutions of Eq. (2.1) with n = n 4 ( x ) 1 2 n ( n 1) 4 2 ( x ) = n 0 1 2 n ( n 1) 00 ; (2.4) are p olynomials of degree n , usually called hyp ergeometric-typ e \discrete" p olynomials y = y n ( x ) p n ( x ). These p olynomials [41 ] are orthogonal in the interval [ a; b 1] with resp ect to the weight function ( x ), i.e., b 1 X x i = a p n ( x i ) p m ( x i ) ( x i ) = nm d 2 n ; x i +1 = x i + 1 ; (2.5) 4
provided that the following condition ( x ) ( x ) x k x = a;b = 0 ; 8 k 0 ; (2.6) holds. The square of the norm of the p olynomial p n ( x ) is given [41] by d 2 n = ( 1) n A nn B 2 n b n 1 X x i = a n ( x i ) = ( 1) n a n B n b n 1 X x i = a n ( x i ) ; ; (2.7) where a n is the leading co ecient of the p olynomial p n ( x ), p n ( x ) = a n x n + b n x n 1 + : : : ; (2.8) and B n is the normalization constant of the Ro drigues-typ e formula p n ( x ) = B n ( x ) 5 n [ n ( x )] ; n = 0 ; 1 ; 2 ; ::: ; : (2.9) where n ( x ) = ( x + n ) n Y m =1 ( x + m ) : (2.10) The use of Eq. (2.9) together with the formula 5 n [ f ( x )] = n X k =0 ( 1) k n k ! f ( x k ) ; (2.11) allows one to obtain [41] an explicit expression for the p olynomials. The symb ol n k ! denotes the binomial co ecients, i.e., n k ! = n ! k !( n k )! : The constants a n and B n are related by a n = B n n 1 Y k =0 0 + 1 2 ( n + k 1) 00 ; a 0 = b 0 : (2.12) For the k dierence derivatives of the p olynomials p n ( x ), it is also fullled [41] a Ro drigues-typ e formula 4 k P n ( x ) = A nk B n k ( x ) 5 n k [ n ( x )] ; (2.13) where A nk = n ! ( n k )! k 1 Y m =0 [ 0 + 1 2 ( n + m 1) 00 ] n ! ( n k )! a k B k ; A n 0 = 1 : (2.14) The most general p olynomial solution of the hyp ergemetric dierence equation (2.1) corresp onds to the case ( x ) = A ( x x 1 )( x x 2 ) ; ( x ) + ( x ) = A ( x x 1 )( x x 2 ) : 5
Without loss of generality we will consider the case A = 1 and x 1 = 0. In this case, the monic p olynomial solutions can b e written as follows [2 , 42] P n ( x ) = ( x 1 ) n ( x 2 ) n ( x 2 x 1 x 2 + n 1) n 3 F 2 n; x ; x 2 x 1 x 2 + n 1 x 1 ; x 2 1 ; (2.15) where the generalized hyp ergeometric function p F q is dened by p F q a 1 ; a 2 ; :::; a p b 1 ; b 2 ; :::; b q 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 ! : (2.16) The four referred families of discrete hyp ergeometric p olynomials are the so-called classical discrete orthogonal p olynomials: Hahn h ; n ( x; N ), Meixner M ; n ( x ), Kravchuk K p n ( x; N ) and Charlier C n ( x ), p olynomials [40, 41 ], whose main data in its monic form are shown in Tables 1-2. They can b e expresed in terms of the hyp ergeometric functions by formulas [41 , Section 2.7 ,p. 49]: h ; n ( x; N ) = (1 N ) n ( + 1) n ( + + n + 1) n 3 F 2 x; + + n + 1 ; n 1 N ; + 1 1 ; (2.17) M ; n ( x ) = ( ) n n ( 1) n 2 F 1 n; x 1 1 ; (2.18) K p n ( x; N ) = ( p ) n N ! ( N n )! 2 F 1 n; x N 1 p ; (2.19) C n ( x ) = ( ) n 2 F 0 n; x 1 : (2.20) These expressions immediately follow from the ab ove representation (2.15) and its dierent limits (more details can b e found in [41 , 42]). 2.2 The rising and falling factorials. The rising factorial p olynomials or Po chhammer symb ols ( x ) n are dened by ( x ) n = x ( x + 1) ( x + n 1) ( x + n ) ( x ) ; (2.21) and they have the prop erties ( x ) n = ( 1) n ( x + 1) ( x n + 1) ; ( x ) n ( x 1) n 1 = n + x 1 x 1 ; ( x ) n + k = ( x ) n ( x + n ) k ; (2.22) as well as the dierence equation analogue to the dierential equation ( x n ) 0 = n x n 1 , 5 ( x ) n = n ( x ) n 1 : (2.23) The falling factorial p olynomials or Stirling p olynomials x [ n ] are p olynomials dened by x [ n ] = x ( x 1) ( x n + 1) ( 1) n ( x ) n = ( x + 1) ( x n + 1) (2.24) They satisfy the equations 4 x [ n ] = nx [ n 1] ; 5 x [ n ] = n ( x 1) [ n 1] : (2.25) 6
Table 1: Main data for monic Hahn and Charlier p olynomials. Hahn Charlier P n ( x ) h ; n ( x ; N ) C n ( x ) ( a; b ) [0 ; N 1] [0 ; 1 ) ( x ) x ( N + x ) x ( x ) ( + 1)( N 1) ( + + 2) x x ( x ) + ( x ) ( x + + 1)( N 1 x ) n n ( n + + + 1) n ( x ) ( N + x )( + x + 1) ( N x )( x + 1) e x ( x + 1) ; 1 ; n N 1 > 0 n ( x ) ( N + x )( n + + x + 1) ( N n x )( x + 1) e x + n ( x + 1) B n ( 1) n ( + + n + 1) n ( 1) n b n n 2 2( + 1)( N 1) + ( n 1)( + 2 N 2) + + 2 n n 2 (2 + n 1) d 2 n n !( + n + 1)( + n + 1)( + + N + n + 1) ( + + 2 n + 1)( N n 1)!( + + n + 1)( + + n + 1) 2 n n ! n Table 2: Main data for monic Meixner and Kravchuk p olynomials. Meixner Kravchuk P n ( x ) M ; n ( x ) K p n ( x ) ( a; b ) [0 ; 1 ) [0 ; N ] ( x ) x x ( x ) ( 1) x + N p x 1 p ( x ) + ( x ) x + p 1 p ( x N ) n (1 ) n n 1 p ( x ) x ( + x ) ( )( x + 1) N ! p x (1 p ) N x ( N + 1 x )( x + 1) > 0 ; 0 < < 1 0 < p < 1, n N n ( x ) x + n ( + x + n ) ( )( x + 1) N ! p x + n (1 p ) N n x ( N + 1 n x )( x + 1) B n 1 ( 1) n ( 1) n (1 p ) n b n n + n 1 2 1 + 1 n [ N p + ( n 1)( 1 2 p )] d 2 n n !( ) n n (1 ) +2 n n ! N ! p n (1 p ) n ( N n )! 7
It is well known that the p olynomials x [ n ] and the x n are closely related one to another by the formulas x [ n ] = n X k =0 s ( k ) n x k ; x n = n X k =0 S ( k ) n x [ k ] (2.26) where s ( k ) n and S ( k ) n are the Stirling numb ers of the rst and second kind resp ectively [1]. Moreover, they satisfy the relations s ( k ) n +1 = s ( k 1) n ns ( k ) n ; S ( k ) n +1 = k S ( k ) n + S ( k 1) n ; 1 k n: (2.27) For the Stirling numb ers of the second kind S ( k ) n one has the closed form [1] S ( k ) n = 1 k ! k X j =0 ( 1) k + j k j ! j n : In particular S (1) n = S ( n ) n = 1 and S ( n 1) n = n 2 ( n 1). 3 Expansions of a p olynomial r m ( x ) . Here we nd the explicit expression of the co ecients c mn in the expansion of an arbitrary p olynomial r m ( x ) in series of the orthogonal discrete hyp ergeometric set f p n g , i.e. r m ( x ) = m X n =0 c mn p n ( x ) ; (3.1) The expansion co ecients will b e given in terms of the p olynomial co ecients ( x ) and ( x ) of the dierence equation (2.1) satised by the p olynomials p n ( x ). Theorem 3.1 The explicit expression of the coecients c mn in the expansion (3.1) is c mn = ( 1) n B n d 2 n b 1 X x = a 5 n r m ( x ) ( x ) n 1 Y k =0 ( x k ) : (3.2) Pro of: Multiplying b oth sides of Eq. (3.1) by p k ( x ) ( x ), and summing b etween a and b 1, the orthogonality relation (2.5) immediately gives c mn = 1 d 2 n b 1 X x = a r m ( x ) p n ( x ) ( x ) dx : (3.3) Use the Ro drigues formula (2.9) for p n ( x ) gives c mn = B n d 2 n b 1 X x = a r m ( x ) 5 n [ n ( x )] dx : (3.4) Using n times the following formula of summation by parts b 1 X x = a f ( x ) 5 g ( x ) = f ( x ) g ( x ) b 1 a 1 b 1 X x = a g ( x 1) 5 f ( x ) ; (3.5) 8
and taking into account the orthogonality condition (2.6) as well as Eqs. (2.6) and (2.23), one obtains c mn = ( 1) n B n d 2 n b 1 X x = a 5 n r m ( x ) n ( x n ) = 1 n ! a n b 1 X x = a 5 n r m ( x ) n ( x n ) b n 1 X x = a n ( x ) : (3.6) Finally, using the expression of n ( x ) as given by (2.10), Eq.(3.6) transforms into the searched Eq. (3.2). Keeping in mind Eqs. (2.7) and (2.10), one observes that Eq. (3.2) allows us to determine the expansion formula (3.1) directly from the expression r m ( x ) and the p olynomial co ecients which characterize the dierence equation veried by the p olynomials f p n g . 3.1 Expansion of the p olynomials ( x ) m , x [ m ] and x m . Let us to apply the ab ove equations (3.1) and (3.2) to the sp ecial cases r m ( x ) = ( x ) m and r m ( x ) = x [ m ] . Since 5 n ( x ) m = m ! ( m n )! ( x ) m n ; and 5 n x [ m ] = m ! ( m n )! ( x n ) [ m n ] ; then, ( x ) m = m X n =0 a mn p n ( x ) ; a mn = ( 1) n m ! ( m n )! B n d 2 n b 1 X x = a ( x ) m n n ( x n ) ; (3.7) x [ m ] = m X n =0 d mn p n ( x ) ; d mn = ( 1) n m ! ( m n )! B n d 2 n b 1 X x = a ( x n ) [ m n ] n ( x n ) : (3.8) To solve the problem x m = m X n =0 e mn p n ( x ) ; (3.9) we notice that x m = m X k =0 S ( k ) m x [ k ] = m X k =0 S ( k ) m k X n =0 d k n p n ( x ) = m X n =0 m X k = n d k n S ( k ) m ! p n ( x ) : Then, e mn = m X k = n d k n S ( k ) m : (3.10) The expressions (3.7)-(3.10) complement and extend similar inversion formulas of classical discrete p olynomials previously and dierently found by various authors for the Stirling p olynomials x [ m ] in a purely analitical way [17] or recurrently [11 , 25, 48 , 55]. We should also mention here that the inversion problems of typ e (3.7) and (3.8) can b e easily solved in the classical case by use of the hyp ergeometric-function representation of these p olynomials [53 ]. 9
which using (2.11) and (2.14) b ecomes into c j 0 n = ( 1) n j ! ( j n )! B n ~ B j ~ B n d 2 n b 1 X x = a j n X k =0 n ( x n ) ~ n ( x n ) j n k ! ( 1) k ~ j ( x n k ) : (5.3) The problem (5.1) is also a particular case of the expansion (3.1); namely when the discrete p olynomial r m ( x ) p osesses the hyp ergeometric character. So, Eq. (5.3) may b e alternatively obtained by the inclusion of that fact in Eq. (3.2). In the case when ~ ( x ) = ( x ) and (~ a; ~ b ) = ( a; b ), the corresp onding connection problem (5.1) involves discrete hyp ergeometric p olynomials orthogonal in the same interval, and the expansion co eecients given by (3.2) reduce as c j 0 n = ( 1) n B n ~ B j ~ A j n d 2 n b 1 X x = a j n X k =0 ( x ) ~ ( x ) j n k ! ( 1) k ~ j ( x n k ) : (5.4) For completeness, let us p oint out that there is another equivalent expression for the connection co ecients c j 0 n which sometimes is very useful. The general p olynomial solution of the equation (2.1) is given by (2.15). Then, the solution for the direct connection problem q j ( x ) = j X k =0 a j k x [ k ] ; (5.5) is given by a j k = ( 1) k ( x 1 ) j ( x 2 ) j ( x 2 x 1 x 2 + j 1) k ( n ) k ( x 1 ) k ( x 2 ) k ( x 2 x 1 x 2 + j 1) j k ! : (5.6) This formula inmediately follows from the identity x [ k ] = ( 1) k ( x ) k and the denition of the generalized hyp ergeometric function (2.16). Let us also remark that sometimes it is b etter to use the combination of the ab ove formula with formula (3.8) wo obtain the searched expansion co ecients. Notice that q j ( x ) = j X k =0 a j k x [ k ] = j X k =0 a j k k X n =0 d k n p n ( x ) = j X n =0 0 @ j n X k =0 a j k + n d k + n n 1 A |{z } c j 0 n p n ( x ) ; (5.7) where a j k + n and d k + n n are given by (5.6) and (3.8), resp ectively. Again here, the co- ecients c j 0 n dep end only on the co ecients of the second order dierence equation of hyp ergeometric typ e (2.1). Finally, let us mention that from Eqs. (5.1)-(5.7) one obtains, as a bypro duct, the solution for the conventional connection problem; i.e., that asso ciated with the four classical orthogonal discrete hyp ergeometric p olynomials (Hahn, Meixner, Kravchuk and Charlier). 5.1 Application to all p ossible pairs of classical p olynomials. In this section we will provide the formulas connecting the dierent families of classical hyp ergeometric discrete p olynomials, which generalize results already obtained by dierent authors using dierent approaches, e.g. [5, 17 , 25, 30, 48], in particular, the most general case involving two Hahn p olynomials is given (see formula (5.17) from b elow) from where, the most general connection formula given by Gasp er [17, Eq. (4.1), pag. 188] is 16
obtained as a particular case. The rst eight cases can b e computed by using (5.3) and the other ones with the help of (5.7). Notice that if we equate b oth expressions (3.8) and (5.7) one can obtain dierent summation formulas involving terminating hyp ergeometric series of the typ e given in the App endix. 5.1.1 Charlier-Charlier From formula (5.4) and using the main data of the Charlier p olynomials (see Table 1) we nd for the connection co ecients b etween the families C j ( x ) = j X n =0 c j 0 n C n ( x ) ; the expression c j 0 n = j n ! ( ) j n : (5.8) 5.1.2 Meixner-Meixner For the Meixner-Meixner problem we have M ; j ( x ) = j X n =0 c j 0 n M ; n ( x ) ; where c j 0 n = j n ! (1 ) n + j n ( j + ) ( + n )( 1) j n j n X k =0 ( 1) k j n k ! k ( n + k + ) ( n + k + ) 2 F 1 n + k + ; j + n + k + ! : Using the transformation formula (A.3), the identity j n k ! = ( 1) k ( n j ) k k ! as well as formula (A.4) we nally obtain c j 0 n = j n ! 1 j n ( + n ) j n 2 F 1 n j ; n + n + (1 ) (1 ) ! : (5.9) In particular, for the sp ecial case = , Eq. (5.9) b ecomes c j 0 n = j n ! ( + n ) j n ( 1)( 1) j n ; The second case corresp onds to = , then (5.9) b ecomes c j 0 n = j n ! 1 j n ( ) j n : 17
5.1.3 Kravchuk-Kravchuk. For the Kravchuk-Kravchuk expansion, K p j ( x; N ) = j X n =0 c j 0 n K q n ( x; M ) ; j min f N ; M g ; the same pro cedure used in the Meixner-Meixner case gives us c j 0 n = j n ! ( M j + 1) j n ( p ) j n 2 F 1 n j ; n N n M q p ! : (5.10) In the particular case p = q its reduces to c j 0 n = j n ! p j n ( N M ) j n ; and for the case M = N c j 0 n = j n ! p q j n ( q p ) j n ( N j + 1) j n : 5.1.4 Meixner-Charlier. In this case we have the expansion M ; j ( x ) = j X n =0 c j 0 n C n ( x ) ; with c j 0 n = j n ! e j n ( j + ) ( 1) j n j n X k =0 ( 1) k ( + n + k ) j n k ! k 1 F 1 j + m + k + ! : If we use the transformation formula (A.6) and the sumation formula (A.7) we nd c j 0 n = j n ! 1 j n ( + n ) j n 1 F 1 n j n + (1 ) ! : (5.11) 5.1.5 Charlier-Meixner. For the Charlier-Meixner expansion C j ( x ) = j X n =0 c j 0 n M ; n ( x ) ; one nds from Eq. (5.4) that c j 0 n = j n ! ( ) j n 2 F 0 n j ; + n { (1 ) ! : (5.12) 18
5.1.6 Meixner-Kravchuk. In the Meixner-Kravchuk case, M ; j ( x ) = j X n =0 c j 0 n K p n ( x; N ) ; j N we nd c j 0 n = j n ! ( n + ) j n 1 j n 2 F 1 n j ; n N n + p ( 1) ! : (5.13) 5.1.7 Kravchuk-Meixner. For the Kravchuk-Meixner connection problem, K p j ( x; N ) = j X n =0 c j 0 n M ; n ( x ) ; j N we have c j 0 n = j n ! ( N + 1 j ) j n ( p ) j n 2 F 1 n j ; n + n N ( 1) p ! : (5.14) 5.1.8 Kravchuk-Charlier. For the Kravchuk-Charlier connection problem, K p j ( x; N ) = j X n =0 c j 0 n C n ( x ) ; j N ; we have c j 0 n = j n ! ( N + 1 j ) j n ( p ) j n 1 F 1 n j n N p ! : (5.15) 5.1.9 Charlier-Kravchuk. For the Charlier-Kravchuk problem, C j ( x ) = j X n =0 c j 0 n K p n ( x; N ) ; j N ; we have c j 0 n = j n ! ( ) j n 2 F 0 n j ; n N { p ! : (5.16) 5.1.10 Hahn-Hahn For the Hahn-Hahn problem, we use Eq. (5.7). A straightforward study of the problem h ; j ( x; M ) = j X n =0 c j 0 n h ; n ( x; N ) ; j min f N 1 ; M 1 g ; 19
allows us to nd c j 0 n = j n ! (1 + n M ) j n (1 + n + ) j n (1 + n + j + + ) j n 4 F 3 n j ; 1 + n N ; n + + 1 ; 1 + j + n + + 1 + n M ; n + + 1 ; 2 n + + + 2 1 ! : (5.17) In the particular case N = M (5.17) reduces to c j 0 n = j n ! (1 + n N ) j n (1 + n + ) j n (1 + n + j + + ) j n 3 F 2 n j ; n + + 1 ; 1 + j + n + + n + + 1 ; 2 n + + + 2 1 ! : 5.1.11 Hahn-Charlier For the Hahn-Charlier problem, h + j ( x; N ) = j X n =0 c j 0 n C n ( x ) ; j M 1 ; we nd that c j 0 n = j n (1 + n N ) j n (1 + n + ) j n (1 + n + j + + ) j n 2 F 2 n j ; 1 + j + n + + 1 + n N n + + 1 ! : (5.18) 5.1.12 Charlier-Hahn For the Charlier-Hahn problem, C j ( x ) = j X n =0 c j 0 n h ; n ( x; N ) ; j N 1 ; we nd that c j 0 n = j n ! ( ) j n 3 F 1 n j ; 1 + n N ; n + + 1 2 n + + + 2 1 ! : (5.19) 5.1.13 Hahn-Meixner For the Hahn-Meixner problem, h ; j ( x; N ) = j X n =0 c j 0 n M ; n ( x ) ; j N 1 ; we nd that c j 0 n = j n ! (1 + n N ) j n (1 + n + ) j n (1 + n + j + + ) j n 3 F 2 n j ; + + j + n + 1 ; + n 1 + n N ; n + + 1 1 ! : (5.20) 20
5.1.14 Meixner-Hahn In the Meixner-Hahn case, M ; j ( x ) = j X n =0 c j 0 n h ; n ( x; N ) ; j N 1 ; we nd that c j 0 n = j n 1 j n ( + n ) j n 3 F 2 n j ; 1 + n N ; n + + 1 + + 2 n + 2 ; + n 1 ! : (5.21) 5.1.15 Hahn-Kravchuk For the Hahn-Kravchuk problem, h ; j ( x; N ) = j X n =0 c j 0 n K p n ( x; M ) ; j min f M 1 ; N g ; we nd that c j 0 n = j n ! (1 + n N ) j n (1 + n + ) j n (1 + n + j + + ) j n 3 F 2 n j ; + + j + n + 1 ; n M 1 + n N ; n + + 1 p ! : (5.22) 5.1.16 Kravchuk-Hahn In the Kravchuk-Hahn case, K p j ( x; M ) = j X n =0 c j 0 n h ; n ( x; N ) ; j min f N 1 ; M g ; we nd that c j 0 n = j n ! p j n ( n M ) j n 3 F 2 n j ; 1 + n N ; n + + 1 + + 2 n + 2 ; n M 1 p ! : (5.23) Some of the ab ove connection formulas have b een previously found in a dierent manner either analytically [17] or recurrently [5, 20 , 25, 30, 48 ]; at times, the recurrence relation for the expansion co ecients may b e solved by symb olic means. Gasp er [17 ] gave the explicit solution of the Charlier-Charlier problem as well as the hyp ergeometric representation of the expansion co ecients of the Meixner-Meixner, Kravchuk-Kravchuk, Hahn- Hahn (with the same interval of orthogonality), Kravchuk-Charlier, Kravchuk-Meixner and Kravchuk-Hahn problems; that is, he only considers the seven connection problems with p ositive co ecients. Lewanowicz [30 ] found recurrently the expansion co ecients for all p ossible pairs of the Charlier, Meixner and Kravchuk families; however, he is only able to lead to an explicit or hyp ergeometric-function solution in the Charlier-Charlier, Meixner-Charlier and Kravchuk-Charlier cases. Ronveaux et al. [48 ] are able to alternatively solve in a recurrent way the Charlier-Charlier, Charlier-Kravchuk, Charlier-Meixner, Meixner-Meixner, Meixner-Charlier and Kravchuk-Kravchuk cases; however, they are only 21
able to nd an explicit solution in the Charlier-Charlier case, and the hyp ergeometricfunction solution in the Kravchuk-Kravchuk case by symb olic means. To this resp ect see also [20 ]. Finally, Ko epf and Schmarsau [25 ] have found with their computer-algebra-based metho d the explicit expression for the co ecients of the connection problem b etween classical discrete p olynomials of the same typ e for some particular choice of the parameters (for example, they consider p olynomials in the same interval of orthogonality or p olynomials with equal parameters) save in the Charlier case, of course, where they obtained the complete solution. It is worth to mention here that the connection co ecients of the aforementioned sixteen cases may b e alternatively obtained by use of general theorems on expansion of generalized hyp ergeometric functions in series of functions of the same kind [29 , 30], such us those describ ed in [36, x 9.1], [16]. Moreover, theorems of similar kind [36 , x 12.4], [54] may b e p otentially used to pro duce recursion formulas for the co ecients of the ab ove expressions. This hyp ergeometric approach is b eing develop ed by S. Lewanowicz [35 ]. Finally, notice that, since the co ecients (5.9)-(5.16) are terminating hyp ergeometric series of the typ e 2 F 1 , 1 F 1 and 2 F 0 , they can b e identied with some classical hyp ergeometric p olynomials, e.g. the Jacobi, Meixner or Kravchuk ( 2 F 1 ), Laguerre ( 1 F 1 ) and Charlier ( 2 F 0 ) p olynomials. Summary and Conclusions We have studied The expansion of a general discrete p olynomial r m ( x ) in series of an arbitrary (alb eit xed) orthogonal set of discrete hyp ergeometric p olynomial f p n g , and The expansion of the pro duct r m ( x ) q j ( x ) in series of the orthogonal set f p n g , where q j ( x ) is any discrete hyp ergeometric p olynomial. The corresp onding expansion co ecients are given in a compact and closed form by means of the co ecients which characterize the second-order dierence equations satised by the involved p olynomial(s) as well as the leading co ecient of their explicit expression. The resulting expressions, which are the main contributions of this work, are given by Eqs. (3.2), (4.9) or (4.4) and (5.3) or (5.7). They allow us to calculate b oth analytically and symb olically the expansion co ecients what is greatly useful to solve very involved mathematical problems, such us, e.g. some of queuing theory, birth and death pro cesses and co ding theory, and to deeply understand some physical phenomena which often require to obtain the matrix representation of quantum-mechanical observables; the determination of the corresp onding matrix elements makes often use of connection and/or linearization formulas of the typ e here considered. Let us p oint out that our results, sp ecially Eq. (4.9) or (4.4), op ens a research avenue to determine practical connection and linearization formulas for arbitrary discrete hyp ergeometric p olynomials. Its extension to generalized linearization expression for pro ducts of any numb er of discrete hyp ergeometric p olynomials of great actual interest [23 , 27 , 31 , 38 ] may b e easily carried out. The mathematical usefulness of these general expressions is illustrated by the explicit of the expansion co ecients of the non-orthogonal families ( x ) m , x [ m ] and x m in terms 22
of each classical discrete family (Hahn, Meixner, Kravchuk and Charlier). Also the co ef- cients of the expansion of the pro ducts ( x ) m q j ( x ), x [ m ] q j ( x ) and x m q j ( x ) are found for an arbitrary discrete hyp ergeometric p olynomial. Furthermore, as a nice bypro duct, the complete solution of the conventional connection problem for all p ossible pairs of classical p olynomials (Hahn, Meixner, Kravchuk and Charlier) is given from the same ro ot in a unifying way. Some sp ecic cases of this problem can b e encountered disp ersely in the literature. Finally, we b elieve that the constructive approach here presented is a complementary, very useful alternative to the metho ds of Markett, Ronveaux et al and Lewanowicz to attack succesfully the long-standing connection and linearization problems of discrete hyp ergeometric p olynomials. The same approach has b een used for hyp ergeometric p olynomials of a continuous variable in b oth connection [50 ] and linearization [6] problems, as well as for q -p olynomials [4 ]. A Some formulas involving hyp ergeometric functions. In this section we will enumerate some relations involving hyp ergeometric functions which were useful in order to obtain the results of the pap er. Formulas involving 2 F 1 . Sp ecial values [1, Chapter 15] 2 F 1 a b c 1 ! = ( c )( c a b ) ( c a )( c b ) ; c 6 = 0 ; 1 ; 2 ; :::; < ( c a b ) > 0 : (A.1) 2 F 1 a b b x ! = (1 x ) a 8 b 2 IR : (A.2) Linear transformation formulas [22 , p. 425] 2 F 1 a b c x ! = (1 x ) a 2 F 1 a c b c x x 1 ! = = (1 x ) c a b 2 F 1 c a c b c x ! : (A.3) A summation formula [22 , Eq. 65.2.2, p. 426] 1 X k =0 ( a ) k ( b ) k k !( c ) k y k 2 F 1 c a c b c + k x ! = (1 x ) a + b c 2 F 1 a b c x + y xy ! : (A.4) Formulas involving 1 F 1 . Sp ecial values [1, Chapter 13] 1 F 1 a a x ! = e x ; 8 a 2 IR : (A.5) 23
Linear transformation formula [22 , p. 431] 1 F 1 a c x ! = e x 1 F 1 c a c x ! : (A.6) A summation formula [22 , Eq. (66.2.5), p. 431] 1 X k =0 ( c a ) k k !( c ) k y k 1 F 1 a c + k x ! = e y 1 F 1 a c x y ! : (A.7) Finally, we want to remind, as we already p ointed out in Section 5.1 , that equating the expressions (3.8) and (5.7) one can obtain dierent summation formulas involving terminating hyp ergeometric series of higher order. Akcnowledgmets This work was done during the stay of the rst author (RAN) at the University of Granada. It has b een partially supp orted by the Europ ean pro ject INTAS-93-219-ext as well as by the Direccion General de Ense ~nanza Sup erior (DGES) of Spain under grant PB 96-0120- C01-01 (RAN) and PB 95-1205 (JSD & RJY) and by the Junta de Andaluca (JSD & RJY) under grant FQM207. We thank Prof. Stanislav Lewanowicz for its helpful discussions and remarks, and also to the unknown referees for their helpful remarks which help us to improve the pap er. References [1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. Dover Publ., New York, 1964. [2] R. Alvarez-No darse, Polinomios hipergeometricos y qpolinomios. Monografas de la Academia de Ciencias de Zaragoza, Spain, (1998) (to app ear). [3] R. Alvarez-No darse and A. Ronveaux, Recurrence relation for connection co ecients b etween q-orthogonal p olynomials of discrete variables in the non-uniform lattice x ( s ) = q 2 s . J. Phys. A: Math. and Gen. 29 (1996), 7165-7175. [4] R. Alvarez-No darse, J. Arves u and J.S. Dehesa, Linearization formulas b etween orthogonal q -p olynomials. (In preparation). [5] I. Area, E. Go doy, A. Ronveaux and A. Zarzo, Minimal recurrence relations for connection co ecients b etween classical orthogonal p olynomials: Discrete case. J. Comp. Appl. Math. (1997) submitted. [6] P.L. Artes, J. Sanchez-Ruiz, A. Martinez-Finkelshtein and J.S. Dehesa, Clebsch- Gordan-typ e expansions for pro ducts of continuous hyp ergeometric p olynomials. Preprint (1997). [7] T. Asahi, Sp ectral theory of dierence equations. Prog. Theor. Phys. Suppl. 36 (1966), 55-96. [8] R. Askey, Orthogonal Polynomials and Special Functions. Regional Conferences in Applied Mathematics. 21 , SIAM, Philadelphia, Pennsylvania, 1975. 24
[9] R. Askey, Orthogonal expansions with p ositive co ecients. Proc. Amer. Math. Soc. 26 , (1965), 1191-1194. [10] R. Askey and G. Gasp er, Convolution structures for Laguerre p olynomials. J. d'Analyse Math. 31 , (1977), 48-68. [11] S. Belmehdi, S. Lewanowicz and A. Ronveaux, Linearization of pro ducts of orthogonal p olynomials of a discrete variable. Applicationes Mathematicae. 24 (1997) (to app ear). [12] N. Bogdanova and V. Gadjokov, Use of orthogonal p olynomials in calibration problems. Computer Phys. Commun. 24 (1981), 225-229. [13] C. Dunkl and D.E. Ramirez, Krawtchouk p olynomials and the symmetrization of hyp ergroups. SIAM J. Math. Anal. 179 (1974), 351-366. [14] C.F. Dunkl, Spherical functions on compact groups and applications to sp ecial functions. Symposia Math. 22 (1977), 145-161. [15] A.R.M. Edmonds, Angular Momentum in Quantum Mechanics (Princeton Univ. Press, Princeton, 1957). [16] J. L. Fields and J. Wimp, Expansion of hyp ergeometric functions in hyp ergeometric functions. Math. Comp 15 (1961), 390{395. [17] G. Gasp er, Pro jection formulas for orthogonal p olynomials of a discrete variable, J. Math. Anal. Appl. 45 (1974), 176-198. [18] G. Gasp er, Positivity and Sp ecial Functions. In Theory and Applications of Special Functions. R. Askey Ed., (Academic Press, N.Y. 1975), 375-433. [19] E. Go doy, A. Ronveaux, A. Zarzo and I. Area, Minimal recurrence relations for connection co ecients b etween classical orthogonal p olynomials: continuous case, J. Comput. Appl. Math (1997). In press. [20] E. Go doy, A. Ronveaux, A. Zarzo and I. Area, On the limit relations b etween classical continuous and discrete orthogonal p olynomials. Preprint (1997). [21] A. Jirari, Second-order Sturm-Liuovil le Dierence Equations and Orthogonal Polynomials . Memoirs of AMS, vol 113 (Amer. Math. So c., Providence, R.I., 1995). [22] E.R. Hansen, A Table of Series and Products. Prentice Hall, New Jersey, 1975. [23] M. N. Hounkonnou, S. Belmedi and A. Ronveaux, Linearization of arbitrary pro ducts of classical orthogonal p olynomials. Preprint (1997). (Submitted) [24] H.F. King and M. Dupuis, Numerical integration using Rys p olynomials. J. Comput. Phys. 21 (1976), 144-165. [25] W. Ko epf and D. Schmersau, Representation of orthogonal p olynomials. Preprint ZIB-Berlin (1997). [26] P.A. Lee, An integral representation and some summation formulas for the Hahn p olynomials. SIAM J. Appl. Math. 19 (1970), 266-272. 25