Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms

Transcription

1 Journal of Integer Sequences, Vol 4 (0, Article Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms Paul Barry School of Science Waterford Institute of Technology Ireland pbarry@witie Abstract Taking the eamples of Legendre and Hermite orthogonal polynomials, we show how to interpret the fact that these orthogonal polynomials are moments of other orthogonal polynomials in terms of their associated Riordan arrays We use these means to calculate the Hankel transforms of the associated polynomial sequences Introduction In this note, we shall re-interpret some of the results of Ismail and Stanton [5, 6] in terms of Riordan arrays These authors give functionals [5] whose moments are the Hermite, Laguerre, and various Meiner families of polynomials In this note, we shall confine ourselves to Legendre and Hermite polynomials Indeed, the types of orthogonal polynomials representable with Riordan arrays is very limited (see below, but it is nevertheless instructive to show that a number of them can be ehibited as moments, again using (parameterized Riordan arrays The essence of the paper is to show that a Riordan array L (either ordinary or eponential defines a family of orthogonal polynomials (via its inverse L if and only if its production matri [8, 9, 0] is tri-diagonal The sequence of moments µ n associated to the family of orthogonal polynomials then appears as the elements of the first column of L In terms of generating functions, this means that if L = (g,f (or L = [g,f], then g( is the generating function of the moment sequence By defining suitable parameterized Riordan arrays, we can ehibit the Legendre and Hermite polynomials as such moment sequences

2 While partly epository in nature, the note assumes a certain familiarity with integer sequences, generating functions, orthogonal polynomials [5,, 30], Riordan arrays [5, 9], production matrices [0, 3], and the Hankel transform of sequences [, 7, 0] We provide background material in this note to give a hopefully coherent narrative Many interesting eamples of sequences and Riordan arrays can be found in Neil Sloane s On-Line Encyclopedia of Integer Sequences (OEIS, [7, 8] Sequences are frequently referred to by their OEIS number For instance, the binomial matri B ( Pascal s triangle is A00738 The plan of the paper is as follows: This Introduction Preliminaries on integer sequences and (ordinary Riordan arrays 3 Orthogonal polynomials and Riordan arrays 4 Eponential Riordan arrays and orthogonal polynomials 5 The Hankel transform of an integer sequence 6 Legendre polynomials 7 Legendre polynomials as moments 8 Hermite polynomials 9 Hermite polynomials as moments 0 Acknowledgements Appendi - The Stieltjes transform of a measure Preliminaries on integer sequences and Riordan arrays For an integer sequence a n, that is, an element of Z N, the power series f( = n=0 a n n is called the ordinary generating function or gf of the sequence a n is thus the coefficient of n in this series We denote this by a n = [ n ]f( For instance, F n = [ n ] is the n-th Fibonacci number A000045, while C n = [ n ] 4 is the n-th Catalan number A00008 The article [] gives eamples of the use of the operator [ n ] We use the notation 0 n = [ n ] for the sequence, 0, 0, 0,, A Thus 0 n = [n = 0] = δ n,0 = ( 0 n Here, we have used the Iverson bracket notation [3], defined by [P] = if the proposition P is true, and [P] = 0 if P is false For a power series f( = n=0 a n n with f(0 = 0 we define the reversion or compositional inverse of f to be the power series f( such that f( f( = We shall sometimes write this as f = Revf

3 For a lower triangular matri (a n,k n,k 0 the row sums give the sequence with general term n k=0 a n,k while the diagonal sums form the sequence with general term n a n k,k k=0 The Riordan group [5, 9], is a set of infinite lower-triangular integer matrices, where each matri is defined by a pair of generating functions g( = + g + g + and f( = f + f + where f 0 [9] We assume in addition that f = in what follows The associated matri is the matri whose i-th column is generated by g(f( i (the first column being indeed by 0 The matri corresponding to the pair g,f is denoted by (g,f or R(g,f The group law is then given by (g,f (h,l = (g,f(h,l = (g(h f,l f The identity for this law is I = (, and the inverse of (g,f is (g,f = (/(g f, f where f is the compositional inverse of f A Riordan array of the form (g(,, where g( is the generating function of the sequence a n, is called the sequence array of the sequence a n Its general term is a n k (or more precisely, [k n]a n k Such arrays are also called Appell arrays as they form the elements of the so-called Appell subgroup If M is the matri (g,f, and a = (a 0,a, is an integer sequence with ordinary generating function A (, then the sequence Ma has ordinary generating function g(a(f( The (infinite matri (g,f can thus be considered to act on the ring of integer sequences Z N by multiplication, where a sequence is regarded as a (infinite column vector We can etend this action to the ring of power series Z[[]] by (g,f : A( (g,f A( = g(a(f( Eample The so-called binomial matri B is the element (, of the Riordan group It has general element ( n k, and hence as an array coincides with Pascal s triangle More generally, B m is the element ( of the Riordan group, with general term ( n k m n k It is easy to show that the inverse B m of B m is given by (, m m +m, +m Eample If a n has generating function g(, then the generating function of the sequence n b n = k=0 a n k is equal to ( g( =, g(, 3

4 while the generating function of the sequence n ( n k d n = a n k k k=0 is equal to ( ( g =, g( The row sums of the matri (g,f have generating function (g,f = g( f( while the diagonal sums of (g,f (sums of left-to-right diagonals in the North East direction have generating function g(/( f( These coincide with the row sums of the generalized Riordan array (g,f: (g,f = g( f( For instance the Fibonacci numbers F n+ are the diagonal sums of the binomial matri B given by (, : ( while they are the row sums of the generalized or stretched [6] Riordan array Each Riordan array (g(,f( has bi-variate generating function given by g( yf( 4, :

5 For instance, the binomial matri B has generating function y = ( + y For a sequence a 0,a,a, with gf g(, the aeration of the sequence is the sequence a 0, 0,a, 0,a, with interpolated zeros Its gf is g( The aeration of a (lower-triangular matri M with general term m i,j is the matri whose general term is given by m r + ( i j i+j,, i j where m r i,j is the i,j-th element of the reversal of M: m r i,j = m i,i j In the case of a Riordan array (or indeed any lower triangular array, the row sums of the aeration are equal to the diagonal sums of the reversal of the original matri Eample 3 The Riordan array (c(,c( is the aeration of (c(,c( A03384 Here c( = 4 is the gf of the Catalan numbers Indeed, the reversal of (c(,c( is the matri with general element ( n + k n k + [k n + ], k n + which begins This is A Then (c(,c( has general element ( n + k + + ( n k, n + n k 5

6 and begins This is A053 Note that (c(,c( = ( +, + We observe that the diagonal sums of the reverse of (c(,c( coincide with the row sums of (c(,c(, and are equal to the central binomial coefficients ( n n A00405 An important feature of Riordan arrays is that they have a number of sequence characterizations [4, 4] The simplest of these is as follows Proposition 4 [4, Theorem, Theorem ] Let D = [d n,k ] be an infinite triangular matri Then D is a Riordan array if and only if there eist two sequences A = [a 0,a,a,] and Z = [z 0,z,z,] with a 0 0, z 0 0 such that d n+,k+ = j=0 a jd n,k+j, (k,n = 0,, d n+,0 = j=0 z jd n,j, (n = 0,, The coefficients a 0,a,a, and z 0,z,z, are called the A-sequence and the Z- sequence of the Riordan array D = (g(, f(, respectively Letting A( be the generating function of the A-sequence and Z( be the generating function of the Z-sequence, we have A( = f(, Z( = f( ( g( f( 3 Orthogonal polynomials and Riordan arrays ( By an orthogonal polynomial sequence (p n ( n 0 we shall understand [5, ] an infinite sequence of polynomials p n (, n 0, of degree n, with real coefficients (often integer coefficients that are mutually orthogonal on an interval [ 0, ] (where 0 = is allowed, as well as =, with respect to a weight function w : [ 0, ] R : 0 p n (p m (w(d = δ nm hn h m, where 0 p n(w(d = h n 6

7 We assume that w is strictly positive on the interval ( 0, Every such sequence obeys a so-called three-term recurrence : p n+ ( = (a n + b n p n ( c n p n ( for coefficients a n, b n and c n that depend on n but not We note that if then a n = k n+ k n, p j ( = k j j + k j j + j = 0,, b n = a n ( ( k n+ k n kn h n, c n = a n k n+ k n k n h n Since the degree of p n ( is n, the coefficient array of the polynomials is a lower triangular (infinite matri In the case of monic orthogonal polynomials (where k n = for all n the diagonal elements of this array will all be In this case, we can write the three-term recurrence as p n+ ( = ( α n p n ( β n p n (, p 0 ( =, p ( = α 0 The moments associated to the orthogonal polynomial sequence are the numbers µ n = 0 n w(d We can find p n (, α n and β n (and in the right circumstances, w( - see the Appendi from a knowledge of these moments To do this, we let n be the Hankel determinant µ i+j n i,j 0 and n, be the same determinant, but with the last row equal to,,, Then p n ( = n, n ( u u More generally, we let H k be the determinant of Hankel type with (i,j-th v v k term µ ui +v j Let Then we have n = H ( 0 n 0 n, n = H n ( 0 n n 0 n n + α n = n n n n, β n = n n n We shall say that a family of polynomials {p n (} n 0 is [3] formally orthogonal, if there eists a linear functional L on polynomials such that p n ( is a polynomial of degree n, L(p n (p m ( = 0 for m n, 7

8 3 L(p n( 0 Consequences of this definition include [3] that L( m p n ( = κ n δ mn, 0 m n, κ n 0, and if q( = n k=0 a kp k (, then a k = L(qp k /L(p k The sequence of numbers µ n = L( n is called the sequence of moments of the family of orthogonal polynomials defined by L Note that where a suitable weight function w( eists, then we can realize the functional L as L(p( = p(w( d R If the family p n ( is an orthogonal family for the functional L, then it is also orthogonal for cl, where c 0 In the sequel, we shall always assume that µ 0 = L(p 0 ( = The following well-known results (the first is the well-known Favard s Theorem, which we essentially reproduce from [8], specify the links between orthogonal polynomials, three term recurrences, and the recurrence coefficients and the gf of the moment sequence of the orthogonal polynomials Theorem 5 [8] (Cf [3], Théorème 9 on pi-4, or [3], Theorem 50 Let (p n ( n 0 be a sequence of monic polynomials, the polynomial p n ( having degree n = 0,, Then the sequence (p n ( is (formally orthogonal if and only if there eist sequences (α n n 0 and (β n n with β n 0 for all n, such that the three-term recurrence p n+ = ( α n p n ( β n p n (, for n, holds, with initial conditions p 0 ( = and p ( = α 0 Theorem 6 [8] (Cf [3], Proposition, (7, on p V-5, or [3], Theorem 5 Let (p n ( n 0 be a sequence of monic polynomials, which is orthogonal with respect to some functional L Let p n+ = ( α n p n ( β n p n (, for n, be the corresponding three-term recurrence which is guaranteed by Favard s theorem Then the generating function g( = µ k k for the moments µ k = L( k satisfies k=0 g( = α 0 α µ 0 β α β β 3 α 3 8

9 Given a family of monic orthogonal polynomials we can write Then we have p n+ ( = ( α n p n ( β n p n (, p 0 ( =, p ( = α 0, p n ( = n+ a n+,k k = ( α n k=0 from which we deduce and n a n,k k k=0 n n a n,k k β n a n,k k k=0 k=0 a n+,0 = α n a n,0 β n a n,0 ( a n+,k = a n,k α n a n,k β n a n,k (3 We note that if α n and β n are constant, equal to α and β, respectively, then the sequence (, α, β, 0, 0, forms an A-sequence for the coefficient array The question immediately arises as to the conditions under which a Riordan array (g,f can be the coefficient array of a family of orthogonal polynomials A partial answer is given by the following proposition Proposition 7 Every Riordan array of the form ( + r + s, + r + s is the coefficient array of a family of monic orthogonal polynomials Proof The array (, +r+s +r+s [7] has a C-sequence C( = n 0 c n n given by and thus + r + s = C(, C( = r s This means that the Riordan array (, +r+s +r+s is determined by the fact that a n+,k = a n,k + i 0 c i a n i,k for n,k = 0,,, where a n, = 0 In the case of (, +r+s +r+s we have Working backwards, this now ensures that a n+,k = a n,k ra n,k sa n,k p n+ ( = ( rp n ( sp n (, where p n ( = n k=0 a n,k n The result now follows from Theorem 5 9

10 We note that in this case the three-term recurrence coefficients α n and β n are constants We have in fact the following proposition (see the net section for information on the Chebyshev polynomials Proposition 8 The Riordan array (, +r+s +r+s is the coefficient array of the modified Chebyshev polynomials of the second kind given by P n ( = ( ( r s n U n, n = 0,,, s Proof The production array of ( +r+s, +r+s is given by r s r s r s r s r s r The result is now a consequence of the article [] by Elouafi, for instance The complete answer can be found by considering the associated production matri of a Riordan arrray, in the following sense The concept of a production matri [8, 9, 0] is a general one, but for this work we find it convenient to review it in the contet of Riordan arrays Thus let P be an infinite matri (most often it will have integer entries Letting r 0 be the row vector r 0 = (, 0, 0, 0,, we define r i = r i P, i Stacking these rows leads to another infinite matri which we denote by A P Then P is said to be the production matri for A P If we let u T = (, 0, 0, 0,,0, then we have and A P = u T u T P u T P ĪA P = A P P 0

11 where Ī = (δ i+,j i,j 0 (where δ is the usual Kronecker symbol: Ī = We have P = A P ĪA P (4 Writing A P = ĪA P, we can write this equation as P = A P A P (5 Note that A P is a beheaded version of A P ; that is, it is A P with the first row removed The production matri P is sometimes [3, 6] called the Stieltjes matri S AP associated to A P Other eamples of the use of production matrices can be found in [], for instance The sequence formed by the row sums of A P often has combinatorial significance and is called the sequence associated to P Its general term a n is given by a n = u T P n e where e = In the contet of Riordan arrays, the production matri associated to a proper Riordan array takes on a special form : Proposition 9 [0, Proposition 3] Let P be an infinite production matri and let A P be the matri induced by P Then A P is an (ordinary Riordan matri if and only if P is of the form ξ 0 α ξ α α ξ α α α P = ξ 3 α 3 α α α 0 0 ξ 4 α 4 α 3 α α α 0 ξ 5 α 5 α 4 α 3 α α where ξ 0 0, α 0 0 Moreover, columns 0 and of the matri P are the Z- and A- sequences, respectively, of the Riordan array A P We recall that we have A( = f(, Z( = ( f( g( f(

12 Eample 0 We consider the Riordan array L where ( λ µ L = + a + b, + a + b The production matri (Stieltjes matri of is given by We note that since L = L = P = S L = ( λ µ + a + b, + a + b a + λ b + µ a b a b a b a b a ( λ µ + a + b, + a + b ( = ( λ µ, + a + b, + a + b we have ( λ µ ( ( L = + a + b, = + a + b + a + b, + a + b λ µ, If we now let L = ( + a, L, + a then [] we obtain that the Stieltjes matri for L is given by λ b + µ b S L = 0 0 b b b 0 We have in fact the following general result [] :,

13 Proposition If L = (g(,f( is a Riordan array and P = S L is tridiagonal, then necessarily a b a b a 0 0 P = S L = 0 0 b a b a b a where and vice-versa f( = Rev and g( = + a + b This leads to the important corollary a b f, Corollary If L = (g(,f( is a Riordan array and P = S L is tridiagonal, with a b a b a 0 0 P = S L = 0 0 b a 0, ( b a b a then L is the coefficient array of the family of orthogonal polynomials p n ( where p 0 ( =, p ( = a, and where b n is the sequence 0,b,b,b,b, p n+ ( = ( ap n ( b n p n (, n, Proof By Favard s theorem, it suffices to show that L defines a family of polynomials {p n (} that obey the above three-term recurrence Now L is lower-triangular and so L is the coefficient array of a family of polynomials p n ( (with the degree of p n ( being n, where We have L 3 = p 0 ( p ( p ( p 3 ( S L L = L L L = L Ī L L = L Ī 3

14 Thus S L L (,,, T = L Ī (,,, T = L (,, 3, T We therefore obtain from which we infer that and or a b a b a b a b a b a p 0 ( p ( p ( p 3 ( p ( = a, = p 0 ( p ( p ( p 3 ( p n+ ( + ap n ( + b n p n ( = p n (, n, p n+ ( = ( ap n ( b n p n (, n, If we now start with a family of orthogonal polynomials {p n (}, p 0 ( =, p ( = a, that for n obey a three-term recurrence p n+ ( = ( ap n ( b n p n (, where b n is the sequence 0,b,b,b,b,, then we can define [3] an associated Riordan array L = (g(,f( by f( = Rev and g( = + a + b a b f Clearly, L is then the coefficient array of the family of polynomials {p n (} Combining these results, we have Theorem 3 A Riordan array L = (g(,f( is the inverse of the coefficient array of a family of orthogonal polynomials if and only if its production matri P = S L is tri-diagonal Proof If L has a tri-diagonal production matri, then by Corollary (, L is the coefficient array of orthogonal polynomials It conversely L is the coefficient array of a family of orthogonal polynomials, then using the fact they these polynomials obey a three-term recurrence and the uniqueness of the Z- and A-sequences, we see using equations ( and (3 along with Proposition 9, that the production matri is tri-diagonal Proposition 4 Let L = (g(, f( be a Riordan array with tri-diagonal production matri S L Then [ n ]g( = L( n, where L is the linear functional that defines the associated family of orthogonal polynomials 4

15 Proof Let L = (l i,j i,j 0 We have [3] n = n l n,i p i ( i=0 Applying L, we get ( n L( n = L l n,i p i ( = i=0 n l n,i L(p i ( = i=0 n l n,i δ i,0 = l n,0 = [ n ]g( i=0 Thus under the conditions of the proposition, by Theorem 6, g( is the gf of the moment sequence µ n = L( n Hence g( has the continued fraction epansion g( = a a b a This can also be established directly To see this, we use the b b a Lemma 5 Let Then f( = Rev + a + b f = a b (f/ Proof By definition, f( is the solution u(, with u(0 = 0, of We find Solving the equation we obtain u + au + bu = u( = a a + (a 4b b v( = av b v, v( = a a + (a 4b b = f( 5

16 Thus we have Now f( = Rev + a + b = g( = immediately implies by the above lemma that a b a b a b f = a b (f/ g( = a a b b a b We note that the elements of the rows of L can be identified with the coefficients of the characteristic polynomials of the successive principal sub-matrices of P Eample 6 We consider the Riordan array ( + a + b, + a + b Then the production matri (Stieltjes matri of the inverse Riordan array ( left-multiplied by the k-th binomial array is given by P = ( ( k k, = k, a + k b a + k b a + k b a + k b a + k b a + k and vice-versa This follows since ( ( + a + b, + a + b + k, + k ( = +a+b, +a+b + (a + k + b, + (a + k + b 6

17 In fact we have the more general result : ( + λ + µ + a + b, ( + a + b + k, + k ( + λ + µ + (a + k + b, + (a + k + b = The inverse of this last matri therefore has production array a + k λ b µ a + k b a + k b a + k b a + k b a + k 4 Eponential Riordan arrays The eponential Riordan group [3, 8, 0], is a set of infinite lower-triangular integer matrices, where each matri is defined by a pair of generating functions g( = g 0 + g + g + and f( = f + f + where g 0 0 and f 0 In what follows, we shall assume g 0 = f = The associated matri is the matri whose i-th column has eponential generating function g(f( i /i! (the first column being indeed by 0 The matri corresponding to the pair f,g is denoted by [g,f] The group law is given by [g,f] [h,l] = [g(h f,l f] The identity for this law is I = [,] and the inverse of [g,f] is [g,f] = [/(g f, f] where f is the compositional inverse of f We use the notation er to denote this group If M is the matri [g,f], and u = (u n n 0 is an integer sequence with eponential generating function U (, then the sequence Mu has eponential generating function g(u(f( Thus the row sums of the array [g,f] have eponential generating function given by g(e f( since the sequence,,, has eponential generating function e As an element of the group of eponential Riordan arrays, the Binomial matri B is given by B = [e,] By the above, the eponential generating function of its row sums is given by e e = e, as epected (e is the egf of n Eample 7 We consider the eponential Riordan array [,], A This array has 7

18 elements and general term [k n] n!, and inverse k! which is the array [,] In particular, we note that the row sums of the inverse, which begin, 0,,, 3, (that is, n, have egf ( ep( This sequence is thus the binomial transform of the sequence with egf ( (which is the sequence starting,, 0, 0, 0, Eample 8 We consider the eponential Riordan array L = [, this matri may be calculated as follows T n,k = n! k k! [n ] ( k = n! k! [n k ]( k = n! ( k k! [n k ] j j=0 = n! ( k + j k! [n k ] j j=0 = n! ( k + n k k! n k = n! ( n k! n k ( j j j ] The general term of Thus its row sums, which have egf ep (, have general term n ( n! n k=0 k! n k This is A0006, the number of sets of lists : the number of partitions of {,,n} into any number of lists, where a list means an ordered subset Its general term is equal to (n!l n (, 8

19 We will use the following [8, 0],important result concerning matrices that are production matrices for eponential Riordan arrays Proposition 9 Let A = (a n,k n,k 0 = [g(,f(] be an eponential Riordan array and let c(y = c 0 + c y + c y +, r(y = r 0 + r y + r y + (7 be two formal power series that that Then r(f( = f ( (8 c(f( = g ( g( (9 (i a n+,0 = i i!c i a n,i (0 (ii a n+,k = r 0 a n,k + i!(c i k + kr i k+ a n,i ( k! or, assuming c k = 0 for k < 0 and r k = 0 for k < 0, a n+,k = k! i k i k i!(c i k + kr i k+ a n,i ( Conversely, starting from the sequences defined by (7, the infinite array (a n,k n,k 0 defined by ( is an eponential Riordan array A consequence of this proposition is that the production matri P = (p i,j i,j 0 for an eponential Riordan array obtained as in the proposition satisfies [0] p i,j = i! j! (c i j + jr i j+ (c = 0 Furthermore, the bivariate eponential function φ P (t,z = n,k p n,k t kzn n! of the matri P is given by Note in particular that we have and φ P (t,z = e tz (c(z + tr(z r( = f ( f(, (3 c( = g ( f( (4 g( f( 9

20 Eample 0 The production matri of [, +] A596 is given by The row sums of L have egf ep ( +, and start,,,,, 9, 5, This is A884 This follows since we have g( = and so g ( = 0, implying that c( = 0, and f( = which gives us f( = and f ( = Thus f ( f( = c( = ( + (+ Hence the bivariate generating function of P is e y ( y, as required Eample The eponential Riordan array A = [, ], or has general term T n,k = n! ( n k! k It is closely related to the Laguerre polynomials Its inverse A is the eponential Riordan array [, ] + + with general term ( n k n! n k!( k This is A0009, the triangle of coefficients of the Laguerre polynomials L n ( The production matri of the matri A = [, + +] is given by This follows since we have g( =, and so g ( =, and f( = ( f( = and f ( = Then + ( c( = g ( f( g( f( = + which yields 0

21 while r( = f ( f( = ( + Thus the bivariate generating function of P is given by e y ( + + ( + y We note that where A = ep(s, S = Eample The eponential Riordan array [ e, ln ( ], or is the coefficient array for the polynomials F 0 ( n,; which are an unsigned version of the Charlier polynomials (of order 0 [, 4, 30] This is A09486 It is equal to [ ( ] [e,], ln, or the product of the binomial array B and the array of (unsigned Stirling numbers of the first kind The production matri of the inverse of this matri is given by

22 which indicates the orthogonal nature of these polynomials We can prove this as follows We have [ ( ] ] e, ln = [e ( e, e Hence g( = e ( e and f( = e We are thus led to the equations r( e = e, c( e = e, with solutions r( =, c( = Thus the bi-variate generating function for the production matri of the inverse array is which is what is required e tz (z + t( z, We can infer the following result from the article [3] by Peart and Woan Proposition 3 If L = [g(,f(] is an eponential Riordan array and P = S L is tridiagonal, then necessarily α β α β α 0 0 P = S L = 0 0 β 3 α β 4 α β 5 α 5 where {α i } i 0 is an arithmetic sequence with common difference α, { β i } is an arithmetic i i sequence with common difference β, and ln(g = (α 0 + β fd, g(0 =, where f is given by and vice-versa f = + αf + βf, f(0 = 0, In the above, we note that α 0 = a where g( is the gf of a 0 =,a,a, We have the important Corollary 4 If L = [g(,f(] is an eponential Riordan array and P = S L is tridiagonal, with α β α β α 0 0 P = S L = 0 0 β 3 α 3 0, β 4 α β 5 α 5

23 then L is the coefficient array of the family of monic orthogonal polynomials p n ( where p 0 ( =, p ( = a = α 0, and p n+ ( = ( α n p n ( β n p n (, n 0 Proof By Favard s theorem, it suffices to show that L defines a family of polynomials {p n (} that obey the above three-term recurrence Now L is lower-triangular and so L is the coefficient array of a family of polynomials p n (, where We have Thus L 3 = p 0 ( p ( p ( p 3 ( S L L = L L L = L Ī L L = L Ī S L L (,,, T = L Ī (,,, T = L (,, 3, T We therefore obtain α β α β α β 3 α β 4 α β 5 α 5 p 0 ( p ( p ( p 3 ( = p 0 ( p ( p ( p 3 (, from which we infer and or p ( = α 0, p n+ ( + α n p n ( + β n p n ( = p n (, n, p n+ ( = ( α n p n ( β n p n ( n If we now start with a family of orthogonal polynomials {p n (} that obeys a three-term recurrence p n+ ( = ( α n p n ( β n p n (, n, with p 0 ( =, p ( = α 0, where {α i } i 0 is an arithmetic sequence with common difference α and { β i } is an arithmetic sequence with common difference β, then we can i i define [3] an associated eponential Riordan array L = [g(,f(] by f = + αf + βf, f(0 = 0, 3

24 and ln(g = (α 0 + β fd, g(0 = Clearly, L is then the coefficient array of the family of polynomials {p n (} Gathering these results, we have Theorem 5 An eponential Riordan array L = [g(,f(] is the inverse of the coefficient array of a family of orthogonal polynomials if and only if its production matri P = S L is tri-diagonal Proposition 6 Let L = [g(, f(] be an eponential Riordan array with tri-diagonal production matri S L Then n![ n ]g( = L( n = µ n, where L is the linear functional that defines the associated family of orthogonal polynomials Proof Let L = (l i,j i,j 0 We have n = n l n,i p i ( i=0 Applying L, we get ( n L( n = L l n,i p i ( = i=0 n l n,i L(p i ( = i=0 n l n,i δ i,0 = l n,0 = n![ n ]g( i=0 Corollary 7 Let L = [g(, f(] be an eponential Riordan array with tri-diagonal production matri S L Then the moments µ n of the associated family of orthogonal polynomials are given by the terms of the first column of L Thus under the conditions of the proposition, g( is the gf of the moment sequence L( n Hence g( has the continued fraction epansion g( = α 0 α β α β β 3 α 3 When we come to study Hermite polynomials, we shall be working with elements of the eponential Appell subgroup By the eponential Appell subgroup AeR of er we understand the set of arrays of the form [f(,] Let A AeR correspond to the sequence (a n n 0, with egf f( Let B AeR correspond to the sequence (b n, with egf g( Then we have 4

25 The row sums of A are the binomial transform of (a n The inverse of A is the sequence array for the sequence with egf f( 3 The product AB is the sequence array for the eponential convolution a b(n = ak b n k with egf f(g( n ( n k=0 k For instance, the row sums of A = [f(,] will have egf given by [f(,] e = f(e = e f( = [e,] f(, which is the egf of the binomial transform of (a n Eample 8 We consider the matri [cosh(,], A9467, with elements The row sums of this matri have egf cosh( ep(, which is the egf of the sequence,,, 4, 8, 6, The inverse matri is [sech(, ], A9879, with entries The row sums of this matri have egf sech( ep( This is A The Hankel transform of an integer sequence The Hankel transform of a given sequence A = {a 0,a,a,} is the sequence of Hankel determinants {h 0,h,h,} where h n = a i+j n i,j=0, ie A = {a n } n N0 h = {h n } n N0 : h n = a 0 a a n a a a n+ a n a n+ a n (5 5

26 The Hankel transform of a sequence a n and its binomial transform are equal In the case that a n has gf g( epressible in the form then we have [8] g( = α 0 α a 0 β α h n = a n+ 0 β n β n βn β n = a n+ 0 β β 3 α 3 n k= β n+ k k (6 Note that this independent from α n ( u u We note that α n and β n are in general not integers Now let H k be the v v k determinant of Hankel type with (i,j-th term µ ui +v j Let ( ( 0 n 0 n n n = H, n = H 0 n n 0 n n + Then we have α n = n n n n, 6 Legendre polynomials We recall that the Legendre polynomials P n ( can be defined by n ( ( n k ( k n + P n ( = ( k k k=0 Their generating function is given by = P n (t n t + t β n = n n (7 n We note that the production matri of the inverse of the coefficient array of these polynomials is given by , n=0

27 which corresponds to the fact that the P n ( satisfy the following three-term recurrence (n + P n+ ( = (n + P n ( np n ( The shifted Legendre polynomials P n ( are defined by They satisfy P n ( = ( n n k=0 ( n k P n ( = P n ( ( n + k k ( k = n k=0 ( n k ( n + k k Their coefficient array begins , and so the first few terms begin,, 6 6 +, , ( k k k We clearly have = ( t + t n=0 P n (t n 7 Legendre polynomials as moments Our goal in this section is to represent the Legendre polynomials as the first column of a Riordan array whose production matri is tri-diagonal We first of all consider the so-called shifted Legendre polynomials We have Proposition 9 The inverse L of the Riordan array ( + r( r + (r + r(r, + (r + r(r has as its first column the shifted Legendre polynomials P n (r The production matri of L is tri-diagonal 7

28 Proof Indeed, standard Riordan array techniques show that we have ( + r( r L = + (r + r(r, + (r + r(r ( =, (r (r + (r + r(r This establishes the first part Now using equations (, with and f( = (r (r +, f( = r(r + (r + r(r, we find that g( = (r +, Z( = (r + r(r, A( = + (r + r(r Hence by Proposition 9 the production matri P = S L of L is given by r r(r r r(r r r(r r r(r r r(r r Corollary 30 The shifted Legendre polynomials are moments of the family of orthogonal polynomials whose coefficient array is given by ( L + r( r = + (r + r(r, + (r + r(r Proof This follows from the above result and Proposition 4 Proposition 3 The Hankel transform of the sequence P n (r is given by n (r(r (n+ Proof From the above, the gf of P n (r is given by (r (r The result now follows from Equation (6 8 r(r r(r (r r(r

29 We note that P n (r = π r(r +r r(r +r n + (r d gives an eplicit moment representation for P n (r Turning now to the Legendre polynomials P n (, we have the following result Proposition 3 The inverse L of the Riordan array ( + r 4 + r + r, + r + r has as its first column the Legendre polynomials P n (r The production matri of L is tridiagonal Proof We have L = = ( + r 4 + r + r, + r + r ( (r r +, ( r r + This proves the first assertion Using equations (9 again, we obtain the following tri-diagonal matri as the production matri of L: r r r r 0 r r 0 0 r 0 4 r r 4 r r 4 Corollary 33 L = ( + r 4 + r + r, + r + r is the coefficient array of a set of orthogonal polynomials for which the Legendre polynomials are moments Proposition 34 The Hankel transform of P n (r is given by (r (n+ n 9

30 Proof From the above, we obtain that the gf of P n (r can be epressed as r r The result now follows from Equation (6 We end this section by noting that P n (r = π r+ r r r gives an eplicit moment representation for P n (r r r 4 r r 4 n + r d 8 Hermite polynomials The Hermite polynomials may be defined as H n ( = n k=0 The generating function for H n ( is given by e t t = ( k ( n k k!(n k! n=0 H n ( tn n! The unitary Hermite polynomials (also called normalized Hermite polynomials are given by He n ( = n Hn ( = n k=0 n! ( n k k! ( + ( n k k n k! Their generating function is given by e t t = n=0 He n ( tn n! We note that the coefficient array of He n is a proper eponential Riordan array, equal to ] [e, 30

31 This array A06635 begins It is the aeration of the alternating sign version of the Bessel coefficient array A00497 The inverse of this matri has production matri , which corresponds to the fact that we have the following three-term recurrence for He n : He n+ ( = He n ( nhe n ( 9 Hermite polynomials as moments Proposition 35 The proper eponential Riordan array ] r L = [e, has as first column the unitary Hermite polynomials He n (r This array has a tri-diagonal production array r Proof The first column of L has generating function e, from which the first assertion follows We now use equations (3 and (4 to calculate the production matri P = S L We have f( =, so that f( = and f ( f( r = = r( Also, g( = e implies that g ( = g((r, and hence c( = g ( f( = r g( f( 3

32 Thus the production array of L is indeed tri-diagonal, beginning r r r r r r We note that L starts r r r r(r 3 3(r 3r 0 0 r 4 6r + 3 4r(r 3 6(r 4r 0 r(r 4 0r + 5 5(r 4 6r + 3 0r(r 3 0(r 5r Thus L = ] r+ [e, is the coefficient array of a set of orthogonal polynomials which have as moments the unitary Hermite polynomials These new orthogonal polynomials satisfy the three-term recurrence with H 0 =, H = r H n+ ( = ( rh n ( + nh n (, Proposition 36 The Hankel transform of the sequence He n (r is given by ( (n+ n k=0 k! Proof By the above, the gf of H n (r is given by r + r + The result now follows from Equation (6 3 r + 4 Turning now to the Hermite polynomials H n (, we have the following result 3

33 Proposition 37 The proper eponential Riordan array [ ] L = e r, has as first column the Hermite polynomials H n (r This array has a tri-diagonal production array Proof The first column of L has generating function e r, from which the first assertion follows Using equations (3 and (4 we find that the production array of L is indeed tri-diagonal, beginning r r r r r r We note that L starts r (r 4r r(r 3 6(r 6r 0 0 4(4r 3 r + 3 6r(r 3 (r 8r 0 8r(4r 4 0r + 5 0(4r 4 r r(r 3 0(r 0r Thus L = [ ] e r+, is the coefficient array of a set of orthogonal polynomials which have as moments the Hermite polynomials These new orthogonal polynomials satisfy the three-term recurrence with H 0 =, H = r H n+ ( = ( rh n ( + nh n (, Proposition 38 The Hankel transform of the sequence H n (r is given by ( (n+ n k=0 k k! Proof By the above, the gf of H n (r is given by r + r + The result now follows from Equation ( r + 6

34 0 Acknowledgements The author would like to thank an anonymous referee for their careful reading and cogent suggestions which have hopefully led to a clearer paper Appendi - The Stieltjes transform of a measure The Stieltjes transform of a measure µ on R is a function G µ defined on C \ R by G µ (z = z t µ(t If f is a bounded continuous function on R, we have f(µ( = lim f(ig µ ( + iyd y 0 + R If µ has compact support, then G µ is holomorphic at infinity and for large z, a n G µ (z = z n+, R R n=0 where a n = R tn µ(t are the moments of the measure If µ(t = dψ(t = ψ (tdt then (Stieltjes-Perron ψ(t ψ(t 0 = t π lim IG µ ( + iyd y 0 + t 0 If now g( is the generating function of a sequence a n, with g( = n=0 a n n, then we can define G(z = ( z g a n = z z n+ By this means, under the right circumstances we can retrieve the density function for the measure that defines the elements a n as moments Eample 39 We let g(z = 4z be the gf of the Catalan numbers Then z G(z = ( ( z g = 4 z Then + y 8 + y y IG µ ( + iy = 4, + y and so we obtain ψ ( = π lim y y 8 + y y 4 + y = (4 π 34 n=0

35 References [] J L Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, 00 [] P Barry, P Rajković, and M Petković, An application of Sobolev orthogonal polynomials to the computation of a special Hankel determinant, in W Gautschi, G Rassias, M Themistocles, eds, Approimation and Computation, Springer, 00 [3] P Barry, On a family of generalized Pascal triangles defined by eponential Riordan arrays, J Integer Seq, 0 (007, Article 0735 [4] G-S Cheon, H Kim, and L W Shapiro, Riordan group involution, Linear Algebra Appl, 48 (008, [5] T S Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, 978 [6] C Corsani, D Merlini, and R Sprugnoli, Left-inversion of combinatorial sums, Discrete Math 80 (998, 07 [7] A Cvetković, P Rajković, and M Ivković, Catalan numbers, the Hankel transform and Fibonacci numbers, J Integer Seq, 5 (00, Article 03 [8] E Deutsch and L Shapiro, Eponential Riordan arrays, lecture notes, Nankai University, 004, available electronically at [9] E Deutsch, L Ferrari, and S Rinaldi, Production matrices, Adv in Appl Math 34 (005, 0 [0] E Deutsch, L Ferrari, and S Rinaldi, Production matrices and Riordan arrays, Ann Comb, 3 (009, [] M Elouafi, A D A Hadj, On the powers and the inverse of a tridiagonal matri, Appl Math Comput, (009, 37 4 [] W Gautschi, Orthogonal Polynomials: Computation and Approimation, Clarendon Press - Oford, 003 [3] I Graham, D E Knuth, and O Patashnik, Concrete Mathematics, Addison Wesley, Reading, MA, 994 [4] Tian-Xiao He, R Sprugnoli, Sequence characterization of Riordan arrays, Discrete Math 009 (009, [5] M E H Ismail and D Stanton, Classical orthogonal polynomials as moments, Can J Math 49 (997,

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37 [33] W-J Woan, Hankel matrices and lattice paths, J Integer Seq, 4 (00, Article 0 00 Mathematics Subject Classification: Primary 4C05; Secondary B83, C0, 5B05, 5B36, 33C45 Keywords: Legendre polynomials, Hermite polynomials, integer sequence, orthogonal polynomials, moments, Riordan array, Hankel determinant, Hankel transform (Concerned with sequences A000007, A000045, A00008, A0006, A00405, A00497, A00738, A009766, A0009, A03384, A053, A06635, A094587, A09486, A596, A884, A9467, A9879, and A55585 Received May 4 00; revised version received December 6 00; January 3 0 Published in Journal of Integer Sequences, February 9 0 Return to Journal of Integer Sequences home page 37