Research Article Determinant Representations of Polynomial Sequences of Riordan Type

Transcription

1 Discrete Mthemtics Volume 213, Article ID , 6 pges Reserch Article Determinnt Representtions of Polynomil Sequences of Riordn Type Sheng-ling Yng nd Si-nn Zheng Deprtment of Applied Mthemtics, Lnzhou University of Technology, Lnzhou 735, Chin Correspondence should be ddressed to Sheng-ling Yng; slyng@lutcn Received 12 November 212; Revised 19 Jnury 213; Accepted 27 Jnury 213 Acdemic Editor: Gi Sng Cheon Copyright 213 S-l Yng nd S-n Zheng This is n open ccess rticle distributed under the Cretive Commons Attribution License, which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited In this pper, using the production mtrix of Riordn rry, we obtin recurrence reltion for polynomil sequence ssocited with the Riordn rry, nd we lso show tht the generl term for the sequence cn be expressed s the chrcteristic polynomil of the principl submtrix of the production mtrix As pplictions, unified determinnt expression for the four kinds of Chebyshev polynomilsisgiven 1 Introduction The concept of Riordn rry is very useful in combintorics The infinite tringles of Pscl, Ctln, Motzkin, nd Schröder re importnt nd meningful exmples of Riordn rry, nd mny others hve been proposed nd developed (see, eg, [1 7]) In the recent literture, Riordn rrys hve ttrcted the ttention of vrious uthors from mny points of view nd mny exmples nd generliztions cn be found (see, eg, [8 12]) A Riordn rry denoted by (g(t), f(t)) is n infinite lower tringulr mtrix such tht its column k (k =, 1, 2, ) hs generting function g(t)f(t) k,whereg(t) = n= g nt n nd f(t) = n= f nt n re forml power series with g =1, f =,ndf 1 =Thtis,thegenerltermofmtrix R = (g(t), f(t)) is r n,k =[t n ]g(t)f(t) k ;here[t n ]h(t) denotes the coefficient of t n in power series h(t) GivenRiordn rry (g(t), f(t)) ndcolumnvectorb=(b,b 1,b 2,) T,the product of (g(t), f(t)) nd B gives column vector whose generting function is g(t) b(f(t)), whereb(t) = n= b nt n If we identify vector with its ordinry generting function, the composition rule cn be rewritten s (g (t),f(t)) b (t) =g(t) b (f (t)) (1) ThispropertyisclledthefundmentltheoremforRiordn rrys nd this leds to the mtrix multipliction for Riordn rrys: (g (t),f(t)) (h (t),l(t)) =(g(t) h(f(t)),l(f(t))) (2) ThesetofllRiordnrrysformsgroupundertheprevios opertion of mtrix multipliction The identity element of the group is (1, t) The inverse element of (g(t), f(t)) is (g (t),f(t)) 1 1 = (, f (t)), (3) g(f (t)) where f(t) is compositionl inverse of f(t) A Riordn rry R = (g(t), f(t)) = (r n,k ) n,k cn be chrcterized by two sequences A=( i ) i nd Z=(z i ) i such tht, for n,k r n+1, =z r n, +z 1 r n,1 +z 2 r n,2 +, (4) r n+1,k+1 = r n,k + 1 r n,k r n,k+2 + If A(t) nd Z(t) re the generting functions for the A- nd Z-sequences, respectively, then it follows tht [9, 13] g (t) = 1 1 tz(f(t)), f(t) =ta(f(t)) (5)

2 2 Discrete Mthemtics If the inverse of R = (g(t), f(t)) is R 1 = (d(t), h(t)), then the A-ndZ-sequences of R re A (t) = t h (t), Z(t) = 1 (1 d(t)) (6) h (t) For n invertible lower tringulr mtrix R,itsproduction mtrix (lso clled its Stieltjes mtrix; see [11, 14]) is the mtrix P=R 1 R,whereRis the mtrix R with its first row removed The production mtrix P cn be chrcterized by the mtrix equlity RP = DR, whered = (δ i+1,j ) i,j (δ is the usul Kronecker delt) Lemm 1 (see [14]) Assume tht R=(r n,k ) is n infinite lower tringulr mtrix with r n,n =ThenRisRiordn rry if nd only if its production mtrix P is of the form z z 1 1 P= ( z z ), (7) z ( ) where (, 1, 2,)is the A-sequence nd (z,z 1,z 2,)is the Z-sequence of the Riordn rry R Definition 2 Let (r n (x)) n be sequence of polynomils where r n (x) is of degree n nd r n (x) = n k= r n,kx k Wesy tht (r n (x)) n is polynomil sequence of Riordn type if the coefficient mtrix (r n,k ) n,k is n element of the Riordn group; tht is, there exists Riordn rry (g(t), f(t)) such tht (r n,k ) n,k = (g(t), f(t)) In this cse, we sy tht (r n (x)) n is the polynomil sequence ssocited with the Riordn rry (g(t), f(t)) Letting r n (x) = n k= r n,kx k, n,theninmtrixform we hve r, r 1, r 1,1 ( r 2, r 2,1 r 2,2 )( r 3, r 3,1 r 3,2 r 3,3 1 x x 2 x 3 r (x) r 1 (x) )=( r 2 (x)) Hence, by using (1), we hve the following lemm r 3 (x) Lemm 3 Let (r n (x)) n be the polynomil sequence ssocited with Riordn rry (g(t), f(t)), nd let r(t, x) = n= r n(x)t n be its generting function Then r (t, x) = (8) g (t) 1 xf(t) (9) In [15], Luzón introduced new nottion T(f g) to represent the Riordn rrys nd gve recurrence reltion for the fmily of polynomils ssocited to Riordn rrys In recent works [16, 17], new definition by mens of determinnt form for Appell polynomils is given Sequences of Appell polynomils re specil of the Sheffer sequences [18]In [19], the uthor obtins determinnt representtion for the Sheffer sequence The im of this work is to propose similr pproch for polynomil sequences of Riordn type, whichrespecilofthegenerlizedsheffersequences[12, 18] A determinnt representtion for polynomil sequences of Riordn type is obtined by using production mtrix of Riordn rry In fct, we will show tht the generl formul for the polynomil sequences of Riordn type cn be expressed s the chrcteristic polynomil of the principl submtrix of the production mtrix As pplictions, determinnt expressions for some clssicl polynomil sequences such s Fiboncci, Pell, nd Chebyshev re derived, nd unified determinnt expression for the four kinds of Chebyshev polynomils [2, 21] is estblished 2 Min Theorem In this section we re going to develop our min theorem Theorem 4 Let R = (g(t), f(t)) be Riordn rry with the Z-sequence (z i ) i nd the A-sequence ( i ) i Let(p n (x)) n be the polynomil sequence ssocited with R 1 Then(p n (x)) n stisfies the recurrence reltion: p n (x) =xp n 1 (x) 1 p n 1 (x) 2 p n 2 (x) (1) n 1 p 1 (x) z n 1 p (x), n > 1, with initil condition p (x) = 1, nd p 1 (x) = x z In generl, for ll n 1, p n (x) is given by the following Hessenberg determinnt: p n (x) = ( 1) n n z x z 1 1 x z x z x z x z n 2 n 2 n 3 n 4 n 5 1 x n 1 z n 1 n 2 n 3 n x (11) Proof Let R = (g(t), f(t)) nd R 1 = (g(t), f(t)) 1 = (d(t), h(t)) Then from definition nd (3), we hve g() = 1 nd d(t) = 1/g(f(t))Henced() = 1 nd p (x) = 1Letting E=(1,x,x 2,) T,thenR 1 E=(p (x), p 1 (x), p 2 (x),) T nd DE = (x, x 2,x 3,) T,whereD = (δ i+1,j )LettingP betheproductionmtrixofr, thenrp = DR, ndpr 1 = R 1 DThusPR 1 E=R 1 DE,ndPR 1 E=xR 1 EInmtrix form, we hve z p (x) xp (x) z 1 1 p 1 (x) xp 1 (x) ( z )( p 2 (x))=( xp 2 (x)) z p 3 (x) xp 3 (x) (12)

3 Discrete Mthemtics 3 Using the block mtrix method, we get The previous mtrix eqution cn be rewritten s p (x) ( z z 1 z 2 z 3 1 )+( 2 1 ) p 1 (x) xp (x) p 2 (x) xp 1 (x) ( p 3 (x))=( xp 2 (x)) p 4 (x) xp 3 (x) (13) 1 x ( 2 1 x p 1 (x) p 2 (x) )( p 3 (x)) x p 4 (x) =p (x) ( x z z 1 z 2 z 3 ) (15) Since Therefore, p 1 (x)=x z,ndforn>1,wehve xp (x) xp 1 (x) ( xp 2 (x))=( xp 3 (x) xp (x) xp 1 (x) )+( xp 2 (x)), xp 3 (x) xp 1 (x) x p 1 (x) p 2 (x) ( xp 2 (x))=( x )( p 3 (x)) x p 4 (x) xp 3 (x) (14) n 1 p 1 (x) p n 2 (x) +( 1 x)p n 1 (x) + p n (x) = z n 1 p (x), or equivlently p n (x) = (x 1 )p n 1 (x) 2 p n 2 (x) n 1 p 1 (x) z n 1 p (x) (16) (17) By pplying the Crmer s rule, we cn work out the unknown p n (x) operting with the first n equtions in (15): p n (x) = n (x z )p (x) 1 x z 1 p (x) 2 1 x z 2 p (x) x z 3 p (x) x z 4 p (x) n 2 n 3 n 4 n 5 n 6 z n 2 p (x) n 1 n 2 n 3 n 4 n 5 1 x z n 1 p (x) (18) =p (x) n x z 1 x z x z x z x z 4 n 2 n 3 n 4 n 5 n 6 z n 2 n 1 n 2 n 3 n 4 n 5 1 x z n 1 After trnsferring the lst column to the first position, n opertion which introduces the fctor ( 1) n 1,thetheorem follows Corollry 5 Let R = (g(t), f(t)) be Riordn rry with production mtrix P Let(p n (x)) n be the polynomil sequence ssocited with R 1 = (g(t), f(t)) 1 Thenp (x) = 1, nd for ll n 1, p n (x) = n det (xi n P n ), (19) where =p,1, P n is the principl submtrix of order n of the production mtrix P nd I n is the identity mtrix of order n

4 4 Discrete Mthemtics 3 Applictions A useful ppliction of Theorem 4 is to find the determinnt expression of well-known sequence We illustrte the idel in the following exmples In the finl prgrph, we will give unified determinnt expression for the four kinds of Chebyshev polynomils Exmple 6 Considering the Riordn rry A = (1/(1 + rt 2 ), t/(1 + rt 2 )), wehve(1/(1 + rt 2 ), t/(1 + rt 2 ))(1/(1 xt)) = 1/(1 xt + rt 2 ) The generting functions of the And Z-sequences of A 1 re A (t) = 1+rt2, Z(t) = rt (2) Let (p n (x)) n bethepolynomilsequencessocited with A = (1/(1 + rt 2 ), t/(1 + rt 2 ))Then(p n (x)) n stisfies the recurrence reltion: p n (x) =xp n 1 (x) rp n 2 (x), n 2, (21) with initil condition p (x) = 1,ndp 1 (x) = x In generl, p n (x) is lso given by the following Hessenberg determinnt: p n (x) = ( 1) n x 1 r x 1 r x 1 r x 1 r x 1 1 x (22) If = 1, r = 1,thenp n (x) becomes the Fiboncci polynomils: F n (x) = ( 1) n x 1 1 x 1 1 x 1 1 x 1 1 x 1 (23) 1 x If =2, r= 1,thenp n (x) gives the Pell polynomils: P n (x) =F n (2x) = ( 1) n 2x 1 1 2x 1 1 2x 1 1 2x 1 1 2x 1 1 2x (24) In cse = 2, r = 1, p n (x) becomes the Chebyshev polynomils of the second kind: U n (x) = ( 1) n 2x 1 1 2x 1 1 2x 1 1 2x 1 1 2x 1 1 2x (25) Exmple 7 Considering the Riordn rry B = ((1 bt 2 )/(1+ rt 2 ), t/(1 + rt 2 )), wehve((1 bt 2 )/(1 + rt 2 ), t/(1 + rt 2 ))(1/(1 xt)) = (1 bt 2 )/(1 xt+rt 2 ) Then the generting functions of the A-ndZ-sequences of B 1 re A (t) = 1+rt2, Z(t) = (b+r) t (26) Let (p n (x)) n bethepolynomilsequencessocited with B = ((1 bt 2 )/(1 + rt 2 ), t/(1 + rt 2 ))Then(p n (x)) n stisfies the recurrence reltion: p n (x) =xp n 1 (x) rp n 2 (x), n 3, (27) with initil condition p (x) = 1, ndp 1 (x) = x, p 2 (x) = 2 x 2 b r In generl, p n (x) is lso given by the following Hessenberg determinnt: p n (x) = ( 1) n x 1 b+r x 1 r x 1 r x 1 r x 1 1 x (28) If =2, b=r=1,thenp n (x) become the Chebyshev polynomils of the first kind 2T n (x) n : 2T n (x) = ( 1) n 2x 1 2 2x 1 1 2x 1 1 2x 1 1 2x 1 1 2x (29)

5 Discrete Mthemtics 5 In cse = 3, b =, r = 2, p n (x) give the Fermt polynomils (see [15]): 3x 1 2 3x 1 2 3x 1 f n (x) = ( 1) n 2 3x 1 2 3x 1 1 3x (3) Exmple 8 Considering the Riordn rry C = ((1 bt)/(1+ rt 2 ), t/(1+rt 2 )),wehve((1 bt)/(1+rt 2 ), t/(1+rt 2 ))(1/(1 xt)) = (1 bt)/(1 xt + rt 2 ) The generting functions of the A-ndZ-sequences of C 1 re A (t) = 1+rt2, Z(t) = b+rt (31) Let (p n (x)) n bethepolynomilsequencessocited with C = ((1 bt)/(1 + rt 2 ), t/(1 + rt 2 )) Then(p n (x)) n stisfies the recurrence reltion: p n (x) =xp n 1 (x) rp n 2 (x), n 3, (32) with initil condition p (x) = 1,ndp 1 (x)=x b, p 2 (x) = 2 x 2 +bx r In generl, p n (x) is lso given by the following Hessenberg determinnt: b x 1 r x 1 r x 1 p n (x) = ( 1) n r x 1 r x 1 1 x (33) If =2, b=1, r=1,thenp n (x) becomes the Chebyshev polynomils of the third kind V n (x): V n (x) = ( 1) n 1 2x 1 1 2x 1 1 2x 1 1 2x 1 1 2x 1 1 2x (34) If =2, b= 1, r=1,thenp n (x) gives the Chebyshev polynomils of the fourth kind W n (x): W n (x) = ( 1) n 1 2x 1 1 2x 1 1 2x 1 1 2x 1 1 2x 1 1 2x (35) Finlly, considering the Riordn rry D = ((1 bt ct 2 )/(1+t 2 ), t/(1+t 2 )),wehve((1 bt ct 2 )/(1+t 2 ), t/(1+ t 2 ))(1/(1 xt)) = (1 bt ct 2 )/(1 xt + t 2 )Thenthe generting functions of the A-ndZ-sequences of D 1 re A (t) = 1+t2 b+(c+1) t, Z(t) = (36) Let (p n (x)) n bethepolynomilsequencessocited with D = ((1 bt ct 2 )/(1 + t 2 ), t/(1 + t 2 ))Then(p n (x)) n stisfies the recurrence reltion: p n (x) =xp n 1 (x) p n 2 (x), n 3, (37) with initil condition p (x) = 1, ndp 1 (x) = x b, p 2 (x) = xp 1 (x) (c + 1)p (x) = 2 x 2 bx c 1For n 1,wehve p n (x) = ( 1) n b x 1 c+1 x 1 1 x 1 1 x 1 1 x 1 1 x (38) Therefore we cn give, now, the following Definition 9 The Chebyshev polynomil of degree n, denoted by C n (x,,b,c), is defined by C n (x,, b, c) = ( 1) n C (x,, b, c) =1, b x 1 c+1 x 1 1 x 1 1 x 1 1 x 1 1 x, n 1, (39)

6 6 Discrete Mthemtics where C n (x,,b,c)is represented by Hessenberg determinnt of order n Note tht C n (x,2,,1) = 2T n (x), C n (x,2,,) = U n (x), C n (x,2,1,) = V n (x), ndc n (x, 2, 1, ) = W n (x) Hence, Definition9 cn be considered s unified form for the four kinds of Chebyshev polynomils Acknowledgments The uthors wish to thnk the editor nd referee for their helpful comments nd suggestions This work ws supported by the Ntionl Nturl Science Foundtion of Chin (Grnt no )ndthenturlsciencefoundtionofgnsu Province (Grnt no 11RJZA49) [17] Y Yng, Determinnt representtions of Appell polynomil sequences, Opertors nd Mtrices, vol 2, no 4, pp , 28 [18] S Romn, The Umbrl Clculus, vol 111 of Pure nd Applied Mthemtics, Acdemic Press, New York, NY, USA, 1984 [19] S l Yng, Recurrence reltions for the Sheffer sequences, Liner Algebr nd its Applictions, vol437,no12,pp , 212 [2] K Aghigh, M Msjed-Jmei, nd M Dehghn, A survey on third nd fourth kind of Chebyshev polynomils nd their pplictions, Applied Mthemtics nd Computtion, vol 199, no1,pp2 12,28 [21] L Comtet, Advnced Combintorics, D Reidel Publishing, Dordrecht, The Netherlnds, 1974 References [1]G-SCheon,HKim,ndLWShpiro, Combintoricsof Riordn rrys with identicl A nd Z sequences, Discrete Mthemtics,vol312,no12-13,pp24 249,212 [2] T X He, Prmetric Ctln numbers nd Ctln tringles, Liner Algebr nd its Applictions, vol438,no3,pp , 213 [3] TXHe,LCHsu,ndPJSShiue, TheSheffergroupnd the Riordn group, Discrete Applied Mthemtics, vol155,no 15, pp , 27 [4] LWShpiro,SGetu,WJWon,ndLCWoodson, The Riordn group, Discrete Applied Mthemtics, vol 34, no 1 3, pp , 1991 [5] LWShpiro, BijectionsndtheRiordngroup, Theoreticl Computer Science,vol37,no2,pp43 413,23 [6] R Sprugnoli, Riordn rrys nd the Abel-Gould identity, Discrete Mthemtics,vol142,no1 3,pp ,1995 [7] R Sprugnoli, Riordn rrys nd combintoril sums, Discrete Mthemtics,vol132,no1 3,pp267 29,1994 [8] T X He nd R Sprugnoli, Sequence chrcteriztion of Riordn rrys, Discrete Mthemtics,vol39,no12,pp , 29 [9]DMerlini,DGRogers,RSprugnoli,ndMCVerri, On some lterntive chrcteriztions of Riordn rrys, Cndin Mthemtics,vol49,no2,pp31 32,1997 [1] E Munrini, Riordn mtrices nd sums of hrmonic numbers, Applicble Anlysis nd Discrete Mthemtics, vol5,no 2, pp 176 2, 211 [11] P Pert nd W J Won, Generting functions vi Hnkel nd Stieltjes mtrices, Integer Sequences, vol 3, rticle 21, no 2, 2 [12] W Wng nd T Wng, Generlized Riordn rrys, Discrete Mthemtics,vol38,no24,pp ,28 [13] E Deutsch, L Ferrri, nd S Rinldi, Production mtrices nd Riordn rrys, Annls of Combintorics,vol13,no1,pp65 85, 29 [14] E Deutsch, L Ferrri, nd S Rinldi, Production mtrices, Advnces in Applied Mthemtics, vol34,no1,pp11 122, 25 [15] A Luzón nd M A Morón, Recurrence reltions for polynomil sequences vi Riordn mtrices, Liner Algebr nd its Applictions,vol433,no7,pp ,21 [16] F A Costbile nd E Longo, A determinntl pproch to Appell polynomils, Computtionl nd Applied Mthemtics,vol234,no5,pp ,21

7 Advnces in Opertions Reserch Advnces in Decision Sciences Applied Mthemtics Algebr Probbility nd Sttistics The Scientific World Journl Interntionl Differentil Equtions Submit your mnuscripts t Interntionl Advnces in Combintorics Mthemticl Physics Complex Anlysis Interntionl Mthemtics nd Mthemticl Sciences Mthemticl Problems in Engineering Mthemtics Discrete Mthemtics Discrete Dynmics in Nture nd Society Function Spces Abstrct nd Applied Anlysis Interntionl Stochstic Anlysis Optimiztion