TRIPLE FACTORIZATION OF SOME RIORDAN MATRICES. Paul Peart* Department of Mathematics, Howard University, Washington, D.C

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1 Pau Peart* Department of Mathematics, Howard University, Washington, D.C. 59 Leon Woodson Department of Mathematicss, Morgan State University, Batimore, MD 39 (Submitted June 99). INTRODUCTION When examining a combinatoria sequence, generating functions are often usefu. That is, if we are interested in anayzing the sequence a, a? a,..., we investigate the forma power series f(x) = a +a x + a x -\. In a recent paper [], techniques are discussed that assist in finding cosed-form expressions for the forma power series for a seect, but arge, set of combinatoria sequences. The methods invove using infinite matrices and the Riordan group. The Riordan group is defined in section of this paper. Each matrix, L, in the Riordan group is associated with a combinatoria sequence and with a matrix, S L, caed the Stietjes matrix. S L is defined in section 3. In this paper, we show that when S L is tridiagona, then L= PCF, where the first factor P is a Pasca-type matrix, the second factor C invoves the generating function for the Cataan numbers, and the third factor F invoves the Fibonacci generating function. The foowing is an exampe: I " o v r i - i o : - 5. r i The matrices in the Riordan group are infinite and ower trianguar. So the exampe shows ony the first seven rows. The first factor on the right is the Pasca matrix. The first coumn in the second factor has C(-x ) as generating function, where is the generating function for the Cataan numbers. The third factor has the Fibonacci numbers in each coumn. See section 6 for further exampes of this tripe factorization. * Partiay supported by NSF grant ]

2 In section, we define the Riordan group R and ist some properties that we use in the proofs of the propositions which are given in section 4. In section 3, we discuss the unique Stietjes matrix S L associated with each L in the Riordan group. In this paper, we concentrate on the subset of R given by R T - {L GR:S L is tridiagona}. In section 5, we derive a recurrence reation for the sequence associated with each member of R T, and we discuss the asymptotic behavior of these sequences. In section 6, we provide two exampes invoving we-known sequences. For each exampe, we give the tripe factorization, the Stietjes matrix, the recurrence reation and asymptotic behavior of the corresponding sequence.. THE RIORDAN GROUP A detaied description of this group is given in []. Here we provide a brief summary. Let M = (rnjj)jj>o be an infinite matrix with eements from C, the set of compex numbers. Let c f (x) be the generating function of the j * coumn ofm That is <*(*) = m n,i*"' We ca M a Riordan matrix if c t (x) - g(x)[/(x)j, where g(x) = + g x + g x +g 3 x 3 + -~, and f(x) = x + f x +f 3 x 3 + -~. In this case, we write M = (g(x), /(*)). We denote by R the set of Riordan matrices. R is a group under matrix mutipication with the foowing properties: ( (*<*), fix)) * (h(x), t(x)) = {g(x)h(f(x)), /(/(x))). (ii) / = (, x) is the identity eement. (iii) The inverse of Mis given by where / is the compositiona inverse of/ M~ = > / ( * ) jk/(*)) (iv) If (a,a y a,...) T is a coumn vector with generating function A(x), then mutipying M = (g(x\ /(*)) on the right by this coumn vector yieds a coumn vector with generating function B(x) = g(x)a(f(x)). 3. STIELTJES MATRI Let L be Riordan and et L be the matrix obtained from L by deeting the first row. For exampe, if I is the identity, we have / =. :. [MAY

3 Observe that L = IL. There exists a unique matrix, S L, such that LS L = L. We ca this matrix the Stietjes matrix of L. Exampe: If Z = x _ -x' -x) then s,= ". 4. PROPOSITIONS Proposition : If L = (g(x), f(x)) is Riordan and S L is tridiagona, then (a) S L=\ h, b b b b. ) / = jc(+ft/ + / ) and g = iff JS^ is as in (a). - Z> - yf ivoq/; Let S L = b y by A b 3 h. With c,(x) the generating function for the z* coumn of L, / >, we have c, = g/"'. By ooking at the first coumn of LS L and L, we obtain b xg + x xgf = g -, i.e., For / >, we obtain from LS L = L, *(*) = -h x- x xf 993] 3

4 c i =x{c i _ +b i c i + i+] c M ). o f = x(+b i f + A i+ f ).: = (ft, - bj)f + ( i+ - J+ )f for a i and; >.. -. ft, = bj and j+ = J+ for a / and; >..". we can take ft = ftj = ft = ft 3 = and A A = yj A^ =..-. / = x( + ft/ + ) - Remark: If 5^ is tridiagona, it has the form in (a) and then either (a) = and/ = -bx and# = -bx - (Z>i +b )x + (bb - Jx or * «A* -bx-j(b -4)x -bx +, (b) * and/ = and g =. x - b x - {xf Proposition : If L = (g, / ) is Riordan ; then S L = S L *+b if and ony ifl = P b L*, where - bx' - bx ft b ft ft 3 3ft 3b ft 4 4ft 3 6ft 4ft (cf. [3], p. 7) Proof: Note that v = ft ft ft ft. = b+. So, Z = P fr Z*^>/L = /P*Z* Conversey, suppose S L =b + $ L *. Then => Z = P*Z* = P'Sp* I* = P b (b+i)l* = bp b L*+P b IL* = bl + P b L* = bl + P b L* S L * = L(b + S L *). P b L* (ft/ + S L,) = MP*!* = P b L* = P b bil*+p b IL* = P b (b + T)L* = P b L* = I(P b L*) = P b L* 4 [MAY

5 Proposition 3: If L = {g, / ) is Riordan and S L = b + s + S b b \ b A A then Z, = P 6^, where Su = s + S. ' o A Proof: This foows immediatey from Proposition. Proposition 4 (PCFFactorization): In Proposition 3, Z^ = C x F es, where Q=(c(^),xc(;ix )) with C() = + [C(JC)] = -V-4x ^, i x ^ v " y ^^^ir^?-* Proof: Let J^ = (ft,/!). Then, from Proposition, we must have, when ^, x -V-4^x a n d /i= ^ & = Ifoc \-sx-{ + S)xf f x = xc(ax ) and g x = -ac-(z + S)x c(jbc )' Now, from section, property, we have But C A, = (c(^vc(^)).( T -^,*) = ( T c</x ) -8-(A ^ / ^ _ + S)x z c(ax z ) \ i -axc(ax z )-Sx z [c(ax z )] c(ax ).\T 6xc(/?x )-<5[xc(/fo: )],xc(ax ) o> - sxc(ax ) - 8x [c{hc )] - c(ax )-zxc(ax )-(A + S)x [c(ax )] o-c(x ) + Ax [c(ax )f=. 993] 5

6 5. RECURRENCE RELATIONS AND ASYMPTOTICS We have proved that when L = (/(x), g{x)) is Riordan and S L is tridiagona with the form s L = h A, b b b \ b. then and /(*) = Jf xn = ~ fa ~^" 4^x " fa + w= /bc =,-v-v/w Using the J.C. P. Mier formua (see Henrici [3]), we obtain for f n the three-term recurrence and for g n the five-term recurrence (#i + )/ + = (n + )Z>/ + ( -»)(ft - 4)/ _, nag = [{In-3)bA-nB]g^ + [(In-3)bB + (3-n)(b -A)A-nC]g n _ + [(n-3)bc + (3-n)(b -4)B]g n _ 3 +[(3-n)(b -A)C]g _ 4 where A-- x, B- x b + x \ -b, C = [ - x bb Q + b^. For the asymptotics, we use the methods described in Wif [4, Ch. 5]. For arge n, we obtain where b > A >. fn (n + \y V n+/ (b+4) I^^n Because there are too many cases to consider, we do not attempt to provide a genera formua for the asymptotic vaue of g n. However, the exampes in section 6 iustrate the techniques invoved. 6. EAMPLES Exampe Big Schroder Numbers: If we take -, b - 3, x = + 8-, and b - b + s -, then j. - 3 x - v x -6x +, - x - v x - 6 x + f= and g =. 4x x g is the generating function for the Big Schroder numbers []. 6 [MAY

7 L = (g,f)with S L and L--p%F W --[^, T^-y {^),^)),[ T L-, x ) [ o : i. 8 4 o ;] i Recurrence Reations: Here A =, B =, C =, (n + )/ + = 3(» +)/ + ( - «)/ _! for n >. / =,/ =. ng _ = 3(n-3)g n _ +(3-n)g n _ 3, (n + )g = 3(n-)g _ + (-n)g _, for Asymptotics: For arge «, fn = [*"]/'(*)' (n + \y3 ' (b+vi)" +/ _ (» +)" 3/ (3 + >/ ) n+/ A 3/4 V^ - 3 / 4 - ^. - x - V x - 6 x + x &-I t^k*'-«r + )»./.-fil + >'"P + j 5 ' «+/ Exampe Leendre Poynomias: We require We take *oo= 4x r - =, b = t, y = + S 4 -tx + t - 3/4 V^ and b n =.b + e = t. 993] 7

8 Tripe Factorization: L = (-ft-vx -ft + ) V* -ft + ' (f -\)y (f -)/4,(f -)/4' t'-i s L = o s-i = t ^ / Recurrence Reations: Here A = ±g-,b= '-^-, C = ± -. (n + )/ + = (#i +)// +( -»)/ _!, for «>. / =,/, =.» g - -(4»-3)rg- _ + (3-»)(+r )^_ +(4»-9)^_3+(3-»)^_ 4, for «>4. g =,g = t,g =^t -^,g 3 =^-^. Asymptotics: We assume that 7 >, so that the roots of x - tx + = are rea. Denote these roots by f and r with f \ < \ r\. We obtain v/ [*"]/(*) = -p ^ix -ft +) - ^ (H + )" 3 7 ^ _ ^ / / -J(r)" + -V^ I f Vr)" (r) - r - w n + V «#" [x"]g(x) = [x"](x i \-/ -ft + > <r)' W+ V^((r) -) REFERENCES. L. Comtet. Advanced Combinatorics. D. Reide Pubishing Company, S. Getu, L. Shapiro, W. Woan, & L. Woodson. "The Riordan Group." Discrete Appied Mathematics 34 (99): P. Henrici. Appied and Computationa Compex Anaysis. Vo. I. New York: Wiey & Sons, H. Wif. Generatingfunctionoogy. New York: Academic Press, 989. AMS Cassification numbers: 5A5, A4 8 [MAY