Meixner-Type Results for Riordan Arrays and Associated Integer Sequences

Transcription

1 Journal of Integer Sequences Vol 3 00 Article 094 Meiner-Type Results for Riordan Arrays and Associated Integer Sequences Paul Barry School of Science Waterford Institute of Technology Ireland pbarry@witie Aoife Hennessy Department of Computing Mathematics and Physics Waterford Institute of Technology Ireland aoifehennessy@gmailcom Abstract We determine which ordinary Riordan arrays are the coefficient arrays of a family of orthogonal polynomials In so doing we are led to introduce a family of polynomials which includes the Boubaker polynomials and a scaled version of the Chebyshev polynomials using the techniques of Riordan arrays We classify these polynomials in terms of the Chebyshev polynomials of the first and second kinds We also eamine the Hankel transforms of sequences associated with the inverse of the polynomial coefficient arrays including the associated moment sequences Introduction With each Riordan array AtBt we can associate a family of polynomials [9] by p n t n AtBt t At Bt n0 The question can then be asked as to what conditions must be satisfied by At and Bt in order to ensure that p n n 0 be a family of orthogonal polynomials This can be

2 considered to be a Meiner-type question [] where the original Meiner result is related to Sheffer sequences ie to eponential generating functions rather than ordinary generating functions: p n t n At epbt n0 In providing an answer to this question we shall introduce a two-parameter family of orthogonal polynomials using Riordan arrays These polynomials are inspired by the well-known Chebyshev polynomials [5] and the more recently introduced so-called Boubaker polynomials [ 5 6] We shall classify these polynomials in terms of the Chebyshev polynomials of the first and second kinds and we shall also eamine properties of sequences related to the inverses of the coefficient arrays of the polynomials under study While partly epository in nature the note assumes a certain familiarity with integer sequences generating functions orthogonal polynomials [4 0 3] Riordan arrays [6 30] production matrices [8 4] and the integer Hankel transform [ 6 7] Many interesting eamples of sequences and Riordan arrays can be found in Neil Sloane s On-Line Encyclopedia of Integer Sequences OEIS [8 9] 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 Riordan arrays 3 Orthogonal polynomials and Riordan arrays 4 Riordan arrays production matrices and orthogonal polynomials 5 Chebyshev polynomials and Riordan arrays 6 The Boubaker polynomials 7 The family of Chebyshev-Boubaker polynomials 8 The inverse matri B 9 A curious relation 0 Acknowledgements Preliminaries on integer sequences and Riordan arrays For an integer sequence a n that is an element of Z N the power series f k0 a k k 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

3 the n-th Fibonacci number A while C n [ n ] 4 is the n-th Catalan number A00008 We use the notation 0 n [ n ] for the sequence A Thus 0 n [n 0] δ n0 0 n Here we have used the Iverson bracket notation [] defined by [P] if the proposition P is true and [P] 0 if P is false For a power series f n0 a n n with f0 0 we define the reversion or compositional inverse of f to be the power series f such that f f We sometimes write f Revf For a lower triangular matri a nk nk 0 the row sums give the sequence with general term n k0 a nk while the diagonal sums form the sequence with general term n a n kk k0 The Riordan group [6 30] 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 [30] The associated matri is the matri whose i-th column is generated by gf i the first column being indeed by 0 The matri corresponding to the pair gf is denoted by gf or Rgf The group law is then given by gf hl gfhl gh fl f The identity for this law is I and the inverse of gf is gf /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 Such arrays are also called Appell arrays as they form the elements of the Appell subgroup If M is the matri gf and a a 0 a is an integer sequence with ordinary generating function A then the sequence Ma has ordinary generating function gaf The infinite matri gf 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 gf : A gf A gaf In [8 9] the notation Tf g is used to denote the Riordan array f Tf g g g Eample The so-called binomial matri B is the element Thus B T of the Riordan group This matri 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 m m m n k It is easy to show that the inverse B m of B m is given by n k 3 +m +m

4 Eample If a n has generating function g then the generating function of the sequence n b n k0 a n k is equal to g g while the generating function of the sequence n n k d n a n k k is equal to k0 g g The row sums of the matri gf have generating function gf g f while the diagonal sums of gf 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 gf: gf 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 using the nomenclature of [5] Riordan array :

5 It is often the case that we work with generalized Riordan arrays where we rela some of the defining conditions above Thus for instance [5] discusses the notion of the stretched Riordan array In this note we shall encounter vertically stretched arrays of the form gh where now f 0 f 0 with f 0 Such arrays are not invertible but we may eplore their left inversion In this contet standard Riordan arrays as described above are called proper Riordan arrays We note for instance that for any proper Riordan array gf its diagonal sums are just the row sums of the vertically stretched array gf and hence have gf g/ f Each Riordan array gf has bi-variate generating function given by g yf 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 0a 0a with interpolated zeros Its gf is g The aeration of a lower-triangular matri M with general term m ij is the matri whose general term is given by m r + i j i+j i j where m r ij is the ij-th element of the reversal of M: m r ij m ii 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 cc A03384 Here c 4 is the gf of the Catalan numbers Indeed the reversal of cc is the matri with general element n + k n k + [k n + ] k n + which begins 5

6 This is A Then c c has general element and begins This is A053 We have n + n k k + n + + n k c c + + We note that the diagonal sums of the reverse of cc 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 [3 ] The simplest of these is as follows Proposition 4 [] Let D [d nk ] 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 such that d n+k+ j0 a jd nk+j kn 0 d n+0 j0 z jd nj 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 and Z denote the generating functions of these sequences respectively we have [0] that f Af g d 00 Zf 6

7 We therefore deduce that and A f Z [ d ] 00 f g f A consequence of this is the following result which was originally established [9] by Luzón: Lemma 5 Let D gf be a Riordan array whose A-sequence respectively Z-sequence have generating functions A and Z Then A Z D d 00 A A 3 Orthogonal polynomials and Riordan arrays By an orthogonal polynomial sequence p n n 0 we shall understand [4 0] 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 wd δ nm hn h m where 0 p nwd h n 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 where a n k n+ k n p j k j j + k j j + j 0 b n a n h i k n+ k n kn h n c n a n k n+ k n k n h n 0 p i wd 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 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 7

8 The moments associated with the orthogonal polynomial sequence are the numbers µ n 0 n wd We can find p n α n and β n from a knowledge of these moments To do this we let n be the Hankel determinant µ i+j n ij 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 ij-th v v k term µ ui +v j Let 0 n 0 n n n H H 0 n 0 n n + Then we have α n n n n n β n n n n Of importance to this study are the following results the first is the well-known Favard s Theorem which we essentially reproduce from [4] Theorem 6 [4] Cf [3 Théorème 9 p I-4] or [33 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 for n holds with initial conditions p 0 and p α 0 Theorem 7 [4] Cf [3 Proposition 7 p V-5] or [33 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 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 k0 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 k0 from which we deduce and n a nk k k0 n n a nk k β n a n k k k0 k0 a n+0 α n a n0 β n a n 0 a n+k a nk α n a nk β n a n k The question immediately arises as to the conditions under which a Riordan array gf can be the coefficient array of a family of orthogonal polynomials A partial answer is given by the following proposition Proposition 8 Every Riordan array of the form + r + s + r + s is the coefficient array of a family of monic orthogonal polynomials Proof By [3] the array +r+s +r+s has a C-sequence C n 0 c n n given by and thus + r + s C C r s Thus the Riordan array +r+s +r+s is determined by the fact that a n+k a nk + i 0 c i d n ik for nk 0 where a n 0 In the case of +r+s +r+s we have Working backwards this now ensures that where p n n k0 a nk n a n+k a nk ra nk sa n k p n+ rp n sp n 9

10 We note that in this case the three-term recurrence coefficients α n and β n are constants We can strengthen this result as follows Proposition 9 Every Riordan array of the form λ µ + r + s + r + s is the coefficient array of a family of monic orthogonal polynomials Proof We have λ µ + r + s λ µ + r + s where λ µ is the array with elements λ µ λ µ λ µ λ µ λ We write and where We now assert that also B b nk A a nk This follows since the fact that tells us that λ µ + r + s + r + s + r + s + r + s + r + s + r + s a n+k a nk ra nk sa n k 3 b n+k b nk rb nk sb n k B λ µ A b n+k a n+k λa nk µa n k b nk a nk λa n k µa n k b nk a nk λa n k µa n k b n k a n k λa n k µa n 3k 0

11 Then using equation 3 and the equivalent equations for a nk and a n k we see that b n+k b nk rb nk sb n k as required Noting that p 0 p r λ p r + λ + λr µ + r s we see that the family of orthogonal polynomials is defined by the α-sequence α 0 r + λrrr and the β-sequence β s + µsss Proposition 0 The elements in the left-most column of L λ µ + r + s + r + s are the moments corresponding to the family of orthogonal polynomials with coefficient array L Proof We let Then Now f is the solution to thus Then Simplifyingwe find that gf λ µ + r + s + r + s L gf f g f u + r + s f s s + s 4r r g f + r f + s f λ f µ f g f s s + µ r + r 4s rs µ + sλ + s µ

12 We now consider the continued fraction This is equivalent to where g r + λ g r s + µ r s r + λ s + µ h h r s h Solving for h and subsequently for g we find that g g f s r We have in fact the following proposition see the net section for information on the Chebyshev polynomials +r+s is the coefficient array of the mod- Proposition The Riordan array +r+s ified Chebyshev polynomials of the second kind given by Proof We have Thus Now P n s n U n r s n 0 t + t U n t n n0 n0 r r U s st + st n st n s r s st + st rt + st t + rt + st + rt + st t Thus t + rt + st + rt + st t r s n U n t n s as required n0

13 For another perspective on this result see [9] λ µ +r+s Corollary The Riordan array is the coefficient array of the generalized Chebyshev polynomials of the second kind given by Q n r s n U n λ r s n U n s µ r s n U n s n 0 s Proof We have +r+s U n [ n ] t + t By the method of coefficients [] we then have and similarly [ n t ] t + t [n ] t + t U n [ n t ] t + t [n ] t + t U n A more complete answer to our original question can be found by considering the associated production matri [7 8] of a Riordan arrray which we look at in the net section 4 Riordan arrays production matrices and orthogonal polynomials The concept of a production matri [7 8] 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 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 then we have and A P u T u T P u T P DA P A P P 3

14 where D δ i+j ij 0 where δ is the usual Kronecker symbol In [4 7] P is called the Stieltjes matri associated with A P The sequence formed by the row sums of A P often has combinatorial significance and is called the sequence associated with 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 with a proper Riordan array takes on a special form : Proposition 3 [8] 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 Moreover columns 0 and of the matri P are the Z- and A-sequences respectively of the Riordan array A P Eample 4 We calculate the production matri of the Riordan array We have and hence Similarly D cc f c f A Z f [ d ] 00 f g [ f ] c [ ] [ ] 4

15 Thus the production matri of D cc is the matri that begins Eample 5 We calculate the production matri of the Riordan array gf c c + + First we have and hence Net since we have f c f A f + g f + + [ Z d ] 00 f g f + [ ] + + [ ] + + Hence the production matri of c c begins We can generalize the last result to give the following important result 5

16 Proposition 6 The Riordan array L where λ µ L + a + b + a + b has production matri Stieltjes matri given by a + λ b + µ a b a 0 0 P S L 0 0 b a b a b a Proof We let gf L By definition of the inverse we have and hence A λ µ + a + b + a + b f Also by definition of the inverse we have + a + b f + a + b λ µ g f + a + b and hence [ Z d ] 00 f g f ] + a + b λ µ [ + a + b + a + b [ + a + b + λ + µ ] a + λ + b + µ Corollary 7 The Riordan array L + a a + b b + a + b + a + b 6

17 has production matri that begins P S L a b a b a b a b a b a Eample 8 We note that since λ µ L + 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 see [3] 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 [3] : Proposition 9 [3] If L gf 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 7

18 where and vice-versa f Rev and g + a + b Of central importance to this note is the following result a b f Proposition 0 If L gf is a Riordan array and P S L is tridiagonal of the form a b a b a 0 0 P S L 0 0 b a 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 0b bbb p n+ ap n b n p n n Proof We are indebted to an anonymous reviewer for the form of the proof that follows The form of the matri P in 4 is equivalent to saying that A + a + b and that Z a + b Now Lemma 5 tells us that if dh is a Riordan array with A and Z the corresponding A-sequence and Z-sequence respectively then A Z dh d 00 A A Note that by assumption d 00 here Thus + a L a + b b T+a a + a + b + a + b +b b +a+b Theorem 5 of [9] now yields the required form of the three-term recurrence for the associated polynomials with coefficient array L That these are orthogonal polynomials then follows by Favard s theorem 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 We consider the Riordan array + a + b + a + b 8

19 Then the production matri Stieltjes matri of the inverse Riordan array left-multiplied by the k-th binomial array k k k is given by P 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 In fact we have the more general result : + λ + µ + a + b + a + b + k + k + λ + µ + a + k + b + a + k + b +a+b +a+b + 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 We note that if L gf is a Riordan array and P S L is tridiagonal of the form given in Eq 4 then the first column of L gives the moment sequence for the weight function associated with the orthogonal polynomials whose coefficient array is L As pointed out by a referee to whom we are indebted for this important observation the main results of the last two sections may be summarized as follows: Proposition Let L d h be a Riordan array Then the following are equivalent: L is the coefficient array of a family of monic orthogonal polynomials d λ µ +r+s and h with s 0 +r+s 9

20 3 The production matri of L is of the form r s r s r 0 0 P S L 0 0 s r s r s r with s 0 4 The bivariate generating function of L is of the form with s 0 λ µ + r t + s Under these circumstances the elements of the left-most column of L are the moments associated with the linear functional that defines the family of orthogonal polynomials 5 Chebyshev polynomials and Riordan arrays We begin this section by recalling some facts about the Chebyshev polynomials of the first and second kind Thus the Chebyshev polynomials of the first kind T n are defined by T n n n n k k k n k n k 5 k0 for n > 0 and T 0 Similarly the Chebyshev polynomials of the second kind U n can be defined by n n k U n k n k 6 k or alternatively as U n n k0 k0 In terms of generating functions we have n+k n k k T n t n n0 + n k k 7 t t + t 0

21 while U n t n n0 t + t The Chebyshev polynomials of the second kind U n which begin have coefficient array This is the generalized Riordan array + + A0537 We note that the coefficient array of the modified Chebyshev polynomials U n / which begin is given by A04930 This is the Riordan array + + The situation with the Chebyshev polynomials of the first kind is more complicated since while the coefficient array of the polynomials T n 0 n which begins

22 is a generalized Riordan array namely + + that of T n which begins is not a Riordan array However the Riordan array + + which begins A0530 A08045 is the coefficient array for the orthogonal polynomials given by 0 n T n / Orthogonal polynomials can also be defined in terms of the three term recurrence that they obey Thus for instance T n T n T n with a similar recurrence for U n Of course we then have U n / U n / U n / for instance This last recurrence corresponds to the fact that the production matri of + + c c is given by Note that many of the above results can also be found in [8]

23 6 The Boubaker polynomials The Boubaker polynomials arose from the discretization of the equations of heat transfer in pyrolysis [ 5 6] starting from an assumed solution of the form N e A H z + ξ m J m t m0 where J m is the m-th order Bessel function of the first kind Upon truncation we get a set of equations with coefficient matri Q zξ 0 ξ Q zξ ξ 0 + ξ Q zξ m ξ m + ξ m in which we recognize the production matri of the Riordan array The Boubaker polynomials B n are defined to be the family of orthogonal polynomials with coefficient array given by We have B 0 and n n k n 4k B n k n k k n k n > 0 8 k0 We also have B n t n n0 + 3t t + t 3

24 +3 The connection to Riordan arrays has already been noted in [9] The matri + + begins and hence we have B 0 B B + B B 4 4 B We can find an epression for the general term of the Boubaker coefficient matri +3 as follows We have n k n k n k n k n+k k + n k where 3 n 6 n n represents the general term of the sequence with gf + 3 Thus the general term of the Boubaker coefficient array is given by n j0 0 j+k j k n j n j n j k + j k 7 The family of Chebyshev-Boubaker polynomials Inspired by the foregoing we now define the Chebyshev-Boubaker polynomials with parameters rs to be the orthogonal polynomials B n ;rs whose coefficient array is the Riordan array B + r + s + + 4

25 That these are orthogonal polynomials is a consequence of the fact that the production array of B is the tridiagonal matri r s We immediately note the factorization It is clear that we have We have B + r + s B n ;rst n n0 B n ; 0 0 U n / rt + st t + t B n ; 0 0 n T n / B n ; 0 3 B n We can characterize B n ;rs in terms of the Chebyshev polynomials as follows Proposition 3 B n ;rs U n / + ru n / + su n / 0 Proof This follows from the definition since + + is the coefficient array for Un / Proposition 4 B n ;rs ru n / + s + U n / + T n / 0 n Proof This follows since + r + s r s Proposition 5 n n k n s + k B n ;rs k n k + ru n / k n k k0 5

26 Proof Indeed the polynomials defined by + s + + are given by n n k n s + k B n ; 0s k n k k n k k0 This can be shown in a similar fashion to Eq 8 We can also use the factorization in Eq 9 to derive another epression for these polynomials The general term of the Riordan array + + is given by n+k a nk n k + n k k while the general term of the array + r + s is given by fn k where fn can be epressed for instance as 0 fn s r + + r s + s n n n Thus the general element of B is given by fn ja jk 0 +r s j0s r+ n j n j j0 +s j k n j j+k + j k k We finish this section by noting that we could have defined a more general family of Chebyshev-Boukaber orthogonal polynomials as follows: Let + r + s B rsαβ + α + β + α + β Then this array is the coefficient array for the polynomials Bn;rs;αβ This is a family of orthogonal polynomials since the production array of B rsαβ is given by α r β s α β α β α β α β α We have Proposition 6 Bn;rs;αβ β n U n α β +r α β n U n +s α β n U n β β 6

27 8 The inverse matri B We recall that the first column of B contains the moment sequence for the weight function defined by the Chebyshev-Boubaker polynomials Bn; r s In this section we shall be interested in studying this sequence including its Hankel transform as well as looking at the row sums and more briefly the diagonal sums of B The inverse of the matri + + corresponding to r s 0 is the much-studied Catalan related matri c c c 4 where c is the generating function of the Catalan numbers C n n+ n n A00008 The inverse of which corresponds to r 0 s is the matri c which again finds many applications It is A08044 Using the theory of Riordan arrays we find that B + r + s c + rc + s c c c + rc + s c c Note also that + r + s B r + s r + s + c c + r + s Thus for instance the generating function of the first column of the inverse matri is c + rc + s c 4 s + r + s + s + r s + r + s 4 s + rs + + r + s 4 7

28 We can find epressions for the general term u n of the sequence given by the first column of B and the general term v n of the row sums B as follows We start with B c c + r + s The first matri has general element n + n k k + n + + n k while the second matri is the sequence Appell array for the generalized Fibonacci numbers F rs n [ n ] + r + s n n i r n i s k i Thus the general term of B is given by i0 T nk n j0 n + j + + n j n j n + j k i0 j k i i r j k i s i 5 Setting k 0 we obtain u n n j0 n + j + + n j n j n + j j i i i0 r j i s i 6 This last equation translates the fact that u n [ n c ] + rc + s c [n ]c c + r + s Note that another epression for u n is given by u n n+ k0 { n n k k } n k j0 n k j j r n k j b j which represents u n as the diagonal sums of the Hadamard product of the reversal of cc and the sequence array of F rs n The general term v n of the row sums of B is given by v n n n k0 j0 n + j + + n j n j n + j k i0 j k i i r j k i s i 8

29 We can deduce from Eq 3 that the weight function for the first column {u n } of the inverse is given by π s 4 s + r + rs + + s + αrsδ s +βrsδ s r 4s s r 4s s «rs+ s «rs+ s for appropriate values of αrs and βrs Thus for r s we find that the terms of the first column have integral representation u n n 4 3 π d i 3 n i i 3 n i while for r s 3 we find that the terms of the first column have integral representation u n n 4 π d+ 3 6 i 4 3 n + 3 i i n 3 i Using the techniques of [ 6] we can prove the following Proposition 7 The Hankel transform of the first column of + r + s + + is given by h n s n We can recouch this result in the following terms: Proposition 8 The moments of the Chebyshev-Boubaker polynomials B n ;rs have Hankel transform equal to s n Recall that the elements of the first column of B are the moments for the density measure associated with the polynomials B n ;rs We also have Proposition 9 The Hankel transform of the row sums of the Riordan matri + r + s + + is given by Proof The gf of the row sums is given by c +rc +s c c h n [ n ] + s r + s + s + r + s s + rs + + r + s The result again follows from the techniques of [ 6] 9

30 We note that the gf of the row sums may be written as + rc + s c c c c + rc + s c c Thus the row sums of the inverse matri are a convolution of [ n ] + rc + s c and the central binomial coefficients n n A00405 or alternatively a convolution of [ n c ] + rc + s c and n n + 0 n Eample 30 The Hankel transforms of the row sums of the inverse matrices and are both given by F n+ In the case of the matri the row sums are epressible as [ n ] 4 5 n n + 0 n n n+ k0 n n F n k+ k k where the first element of the convolution is n A09865n Note that these constituent sequences have Hankel transforms of n and respectively Alternatively the row sums are given by [ n ] n 5 n In this case the Hankel transforms of the constituent elements of the convolution are given by and the all s sequence For the case of the matri n n + 0 n we note that c + 3c + c

31 and so the row sums of the inverse in this case are simply given by n k 3 n k k + 0 k k0 We finish this section by noting that the diagonal sums of B are also of interest They have generating function For instance the diagonal sums of + r s + r 4 s + r + s + r + s + + which begin have as Hankel transform the -period sequence with gf which begins Similarly the diagonal sums of the matri which begin have as Hankel transform the sequence with gf n cos πn πn + 3n + sin and general term We can conjecture that the Hankel transform of the diagonal sums of B in the general case is given by [ n ] + s s 3 + r A curious relation The third column of the Boubaker coefficient matri t n n j0 +3 has general term j j + j n j n j n j 3

32 This sequence begins Now the sequence t n+ is therefore given by This is A3386 with general term n n The interested reader may wish to verify that t n+ n + π R 4 d here R returns the real part of a comple number 0 Acknowledgements The authors would like to thank the anonymous referees for their careful reading and cogent suggestions which have hopefully led to a clearer paper We have also been happy to change the working title of the paper at the suggestion of a referee as the new title more properly captures the thrust of the work References [] P Barry P Rajkovic and M Petkovic An application of Sobolev orthogonal polynomials to the computation of a special Hankel determinant in W Gautschi G Rassias and M Themistocles eds Approimation and Computation Springer 00 [] K Boubaker A Chaouachi M Amlouk and H Bouzouita Enhancement of pyrolysis spray dispersal performance using thermal time-response to precursor uniform deposition Eur Phys J AP [3] G-S Cheon H Kim and L W Shapiro Riordan group involution Linear Algebra Appl [4] T S Chihara An Introduction to Orthogonal Polynomials Gordon and Breach New York [5] C Corsani D Merlini and R Sprugnoli Left-inversion of combinatorial sums Discrete Math [6] A Cvetković P Rajković and M Ivković Catalan numbers the Hankel transform and Fibonacci numbers J Integer Seq 5 00 Article 03 [7] E Deutsch L Ferrari and S Rinaldi Production Matrices Advances in Appl Math

33 [8] E Deutsch L Ferrari and S Rinaldi Production matrices and Riordan arrays preprint February 007 [9] M Elouafi and A D A Hadj On the powers and the inverse of a tridiagonal matri Applied Math Comput [0] W Gautschi Orthogonal Polynomials: Computation and Approimation Clarendon Press Oford [] I Graham D E Knuth and O Patashnik Concrete Mathematics Addison Wesley Reading MA [] Tian-Xiao He and R Sprugnoli Sequence characterization of Riordan arrays Discrete Math [3] S-T Jin A characterization of the Riordan Bell subgroup by C-sequences Korean J Math [4] C Krattenthaler Advanced determinant calculus Sém Lotharingien Combin Article B4q Available electronically at [5] H Labiadh and K M Boubaker A Sturm-Liouville shaped characteristic differential equation as a guide to establish a quasi-polynomial epression to the Boubaker polynomials Differ Uravn Protsessy Upr [6] H Labiadh M Dada O B Awojoyogbe K B Ben Mahmoud and A Bannour Establishment of an ordinary generating function and a Christoffel-Darbou type firstorder differential equation for the heat equation related Boubaker-Turki polynomials Differ Uravn Protsessy Upr [7] J W Layman The Hankel transform and some of its properties J Integer Seq 4 00 Article 05 [8] A Luzón Iterative processes related to Riordan arrays: The reciprocation and the inversion of power series preprint [9] A Luzón and M A Morón Recurrence relations for polynomial sequences via Riordan matrices Linear Alg Appl [0] D Merlini D G Rogers R Sprugnoli and M C Verri On some alternative characterizations of Riordan arrays Canad J Math [] D Merlini R Sprugnoli and M C Verri The method of coefficients Amer Math Monthly [] J Meiner Orthogonale Polynomsysteme mit einer besonderen Gestalt der Erzeugenden Funktion J London Math Soc

34 [3] P Peart and L Woodson Triple factorization of some Riordan matrices Fibonacci Quart [4] P Peart and W-J Woan Generating functions via Hankel and Stieltjes matrices J Integer Seq Article 00 [5] T J Rivlin Chebyshev Polynomials: From Approimation Theory to Algebra and Number Theory Wiley-Interscience nd edition 990 [6] L W Shapiro S Getu W-J Woan and L C Woodson The Riordan group Discr Appl Math [7] L W Shapiro Bijections and the Riordan group Theoret Comput Sci [8] N J A Sloane The On-Line Encyclopedia of Integer Sequences Published electronically at njas/sequences/ 00 [9] N J A Sloane The on-line encyclopedia of integer sequences Notices Amer Math Soc [30] R Sprugnoli Riordan arrays and combinatorial sums Discrete Math [3] G Szegö Orthogonal Polynomials 4th edition Amer Math Soc 975 [3] X J Viennot Une théorie combinatoire des polynômes orthogonau générau UQAM Montreal Quebec 983 [33] H S Wall Analytic Theory of Continued Fractions AMS Chelsea Publishing 983 [34] W-J Woan Hankel matrices and lattice paths J Integer Seq 4 00 Article 0 00 Mathematics Subject Classification: Primary 4C05; Secondary B83 33C45 B39 C0 5B05 5B36 Keywords: Chebyshev polynomials Boubaker polynomials integer sequence orthogonal polynomials Riordan array production matri Hankel determinant Hankel transform Concerned with sequences A A A00008 A00405 A00738 A A03384 A04930 A0537 A0530 A053 A09865 A08044 A08045 and A3386 Received May 4 00; revised version received September 8 00; October 4 00 Published in Journal of Integer Sequences December 6 00 Return to Journal of Integer Sequences home page 34