We obtain an existing 2004 result of J. McLaughlin which gives explicit entries for a general dimension 2 matrix raised to an arbitrary power. Our formulation employs so called Catalan polynomials related to the crucial parameter of McLaughlin’s statement, and is a new one running along a different line of argument

Larcombe, Peter J.

We obtain an existing 2004 result of J. McLaughlin which gives explicit entries for a general dimension 2 matrix raised to an arbitrary power. Our formulation employs so called Catalan polynomials related to the crucial parameter of McLaughlin’s statement, and is a new one running along a different line of argument.We obtain an existing 2004 result of J. McLaughlin which gives
explicit entries for a general dimension 2 matrix raised to an
arbitrary power. Our formulation employs so called Catalan
polynomials related to the crucial parameter of McLaughlin’s
statement, and is a new one running along a different line of
argument

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A New Formulation of a Result byMcLaughlin for an ArbitraryDimension 2 Matrix PowerPeter J. LarcombeDepartment of Computing and MathematicsCollege of Engineering and TechnologyUniversity of DerbyKedleston Road, Derby DE22 1GB, U.K.{P.J.Larcombe@derby.ac.uk}AbstractWe obtain an existing 2004 result of J. McLaughlin which givesexplicit entries for a general dimension 2 matrix raised to anarbitrary power. Our formulation employs so called Catalanpolynomials related to the crucial parameter of McLaughlin’sstatement, and is a new one running along a different line ofargument.1 Introduction1.1 BackgroundLet M be a general 2× 2 matrixM = M(A,B,C,D) =(A BC D), (1)for which it is assumed that each of A,B,C,D is non-zero and, further, thatthe matrix M has both a non-zero trace T = T (A,D) = A + D and non-zero determinant M = M(A,B,C,D) = |M| = AD − BC. With α1 = A,β1 = B, γ1 = C, δ1 = D, suppose the matrix has nth power (n ≥ 1)Mn =(αn βnγn δn), (2)1where αn = αn(A,B,C,D), . . . , δn = δn(A,B,C,D). A theorem, publishedrecently by the author [1], states that the ratio βn/γn (= B/C) is a quantityindependent of matrix power n. It is noted therein, where various proofsare presented, that this invariance property is delivered trivially by a resultof McLaughlin [2, Theorem 1, p.3] who, on defining a parameteryn = yn(A,B,C,D) = yn(T (A,D),M(A,B,C,D))=b 12nc∑i=0(n− ii)Tn−2i(−M)i, (3)proved inductively that, for n ≥ 1,Mn(A,B,C,D) =(yn −Dyn−1 Byn−1Cyn−1 yn −Ayn−1); (4)the anti-diagonals ratio of Mn is immediate as Byn−1/Cyn−1 = B/C, and(4) was used to formulate a number of combinatorial identities by him.1.2 This PaperIn this paper we give an independent proof of (4) via a different routebased on the notion of the so called Catalan polynomial closely relatedto McLaughlin’s parameter, ending with some remarks in order to placethe work in context. Its broader origins lie in a condition found for anti-diagonals product invariance across powers of 2×2 matrix sets [3] in whichA = A(x), B = B(x) and C = C(x) are drawn from Z[x] and characterise aparticular class of polynomial families (being those functional coefficients ofa quadratic equation satisfied by the ordinary generating function of a suiteof associated integer sequences; see Remark 1 later). As an isolated resultof linear algebra the invariance of the anti-diagonals ratio of an arbitrary2-square matrix with respect to its power would seem little known, which issurprising.The general (n + 1)th Catalan polynomial Pn(x) is, for n ≥ 0, definedasPn(x) =b 12nc∑i=0(n− ii)(−x)i, (5)with the connecting relationyn(A,B,C,D) = yn(T (A,D),M(A,B,C,D)) = TnPn(M/T2) (6)a simple one to which we will obviously make appeal. The first few poly-nomials are P0(x) = P1(x) = 1, P2(x) = 1 − x, P3(x) = 1 − 2x, P4(x) =1 − 3x + x2, P5(x) = 1 − 4x + 3x2, P6(x) = 1 − 5x + 6x2 − x3, P7(x) =1− 6x+ 10x2 − 4x3, and so on, and the crucial result enabling our formu-lation of (4) is that, for any x, y [4, (21), p.142],(1 xy 0)n=(Pn(−xy) xPn−1(−xy)yPn−1(−xy) xyPn−2(−xy)); (7)evidently (7) gives directly the said invariance result (the statement of whichis unaffected) in the particular D = 0 case for M.2 The ProofWe begin by defining matricesQ(C,D) =(1 −D/C0 1)(P1)andR(A,B,C,D) =(1 −M/CTC/T 0), (P2)so that, conveniently,TQ(C,D)R(A,B,C,D)Q−1(C,D) = M(A,B,C,D), (P3)and in turnMn(A,B,C,D) = TnQ(C,D)Rn(A,B,C,D)Q−1(C,D). (P4)Equation (7), with x = −M/CT , y = C/T (so that −xy = M/T 2 =r(T,M), say), allows us to write down the nth power of R asRn(A,B,C,D) =(Pn(r) − MCT Pn−1(r)CT Pn−1(r) −rPn−2(r)), (P5)whence (P4) readsMn(A,B,C,D) = Tn(M ′11(A,B,C,D;n) M′12(A,B,C,D;n)M ′21(A,B,C,D;n) M′22(A,B,C,D;n)), (P6)say, and where, after a little algebra, the precise entries of Mn in (P6) areobtained through (P1),(P5) and matched with those of (4) as now shown.Firstly, we find thatTnM ′11(A,B,C,D;n) = Tn(Pn(r)−DTPn−1(r))= TnPn(r)−DTn−1Pn−1(r)= yn −Dyn−1,TnM ′21(A,B,C,D;n) = Tn · CTPn−1(r)= CTn−1Pn−1(r)= Cyn−1, (P7)having used (6). To deal with the remaining entries, however (in order toarrive at (4) as required), we need a further sub-result. For n ≥ 2 (givenP0(x) = P1(x) = 1), the Catalan polynomials satisfy the linear order 2recurrence [5, (69), p.17]Pn(x) = Pn−1(x)− xPn−2(x), (P8)from which, incidentally, the closed formPn(x) =12n+1(1 +√1− 4x )n+1 − (1−√1− 4x )n+1√1− 4x(P9)is readily established1 and by (6) infersyn =12(T2)n(1 +√1− 4M/T 2 )n+1 − (1−√1− 4M/T 2 )n+1√1− 4M/T 2, (P10)which is absent from McLaughlin’s work [2]. Setting x = r = M/T 2 in (P8)and multiplying throughout by Tn, we have the corresponding recursionyn = Tyn−1 −Myn−2 (P11)for McLaughlin’s parameter, which we deploy as seen below (note that in [2](P11) is necessarily derived so as to actually complete the inductive proofof (4), and holds for n ≥ 2 with y0 = 1, y1 = T ). Continuing,TnM ′22(A,B,C,D;n) = Tn(DTPn−1(r)− rPn−2(r))= DTn−1Pn−1(r)−MTn−2Pn−2(r)1We remark that the 2008 article [5] is the first one in which these polynomialsmade a named appearance in the literature, having been discovered in a study of pro-cesses underlying the sequential production of what has been termed iterated generatingfunctions—they were found to give rise to a simple mechanism (an order 1 recursivescheme) to iteratively deliver polynomials which act as ordinary generating functions foran increasing term count of a pre-specified (and finite) Catalan number sequence; equa-tions (P8),(P9), and other properties of the Catalan polynomials, are given in Section 5therein (see also Remark 2(a) here).= Dyn−1 −Myn−2= Dyn−1 + (yn − Tyn−1)= Dyn−1 + yn − (A+D)yn−1= yn −Ayn−1, (P12)and, finally,TnM ′12(A,B,C,D;n)= Tn(DCPn(r)−1CT(M +D2)Pn−1(r) +DCrPn−2(r))=1C[DTnPn(r)− (M +D2)Tn−1Pn−1(r) +DMTn−2Pn−2(r)]=1C[Dyn − (M +D2)yn−1 +DMyn−2]=1C[Dyn − (M +D2)yn−1 +D(Tyn−1 − yn)]=1C[DT − (M +D2)]yn−1=1C[D(A+D)− (AD −BC +D2)]yn−1=1C[BC]yn−1= Byn−1. (P13)This ends the proof.2We conclude the paper with some remarks which place it in a better contextfor the reader.3 Some Additional RemarksRemark 1 Being in a position to do so, then for completeness and interestwe recover an unusual result from an earlier paper. Let A(x), B(x), C(x) ∈Z[x], and suppose the (ordinary) generating function T (x) of a sequence ofintegers satisfies a general quadratic governing equation0 = A(x)T 2(x) +B(x)T (x) + C(x). (8)The functional coefficients A(x), B(x), C(x) can be considered to give riseto a family of associated polynomials α0(x), α1(x), α2(x), . . . , defined asαn(x) = αn(A(x), B(x), C(x))= (1, 0)(−B(x) A(x)−C(x) 0)n(10), n ≥ 0, (9)as alluded to earlier. Now, setting D = 0 in (4) then(1, 0)(A BC 0)n(10)= (1, 0)Mn(A,B,C, 0)(10)= yn(A,B,C, 0)= AnPn(−BC/A2), (10)by (6). Thus, from comparison with (9), we haveαn(A(x), B(x), C(x)) = [−B(x)]nPn(A(x)C(x)B2(x)), (11)a curious relation first given in [6, Theorem 3, p.21].It is felt that Remarks 2(a),(b) which follow are important ones, as theyclarify the relationship between the Catalan polynomials and what are un-derstood to be Fibonacci polynomials, addressing also an ambiguity withinthe definition of the latter which from the literature would appear to beunresolved still (the author would like to thank a referee for motivating thisaddition to the paper). It would be fair to say that the connection betweenCatalan polynomials and a matrix power (7) is one that has proven itselfto be surprisingly rich in its application, of which that here is but one par-ticular instance.Remark 2(a) One version of Fibonacci polynomials fn(x) (n ≥ 0) com-prises, with f0(x) = f1(x) = 1, those given by the recursion fn+1(x) =fn(x) +xfn−1(x) (n ≥ 1)—see the ordered listing of coefficients as O.E.I.S.Sequence No. A011973. They have been known for many years, appearingwithin, for example, the evaluation of the exponentiated matrix(1 x1 0)ina 1919/20 paper by E.E. Jacobsthal [7, pp.44,45]—indeed Koshy, in his2001 text, refers to them as Jacobsthal polynomials [8, Chapter 39]. Theyhave the property that, for n ≥ 0, fn(x) = Pn(−x) (the Catalan polyno-mial coefficients are found as Sequence No. A115139, and {Pn(−1)}∞n=0 ={fn(1)}∞n=0 = {1, 1, 2, 3, 5, . . .} delivers the Fibonacci sequence), which ofcourse means that the identity (7) could be re-cast in terms of these poly-nomials and McLaughlin’s result forced through accordingly.Remark 2(b) It is worth making the point that the polynomials describedabove are not the same as what seem to be known in more modern timesas Fibonacci polynomials also, but which latter satisfy a slightly differentrecurrence (i.e., fn+1(x) = xfn(x) + fn−1(x)) and are (after possibly 0to start) 1, x, x2 + 1, x3 + 2x, x4 + 3x2 + 1, . . . , only reproducing SequenceNo. A011973 when coefficients are read in descending power order (ignoringabsent powers); these polynomials have their own properties, naturally, andseem to have become properly visible in the literature as named ones fromthe mid-late 1960s onwards [9-11].2Remark 3 Powers of arbitrary 2 × 2 matrices have been studied in thepast (see, e.g., [13,14]), often using an eigenvalue approach which limits theformat of results and so their usefulness as far as we are concerned. Anextension of McLaughlin’s work was made by Belbachir and Bencherif inwhich, for m ≥ 2, an explicit expression for the general term of a degreem linear recurrence—with coefficients drawn from a unitary commutativering—was found, allowing the generalisation of McLaughlin’s result to ac-commodate powers of a square order m matrix (although the structure ofthe result [15, Theorem 4, pp.10-11] is not particularly illuminating); this inturn led to some identities involving Fibonacci and Stirling numbers, andthe derivation of various other combinatorial relations. McLaughlin andSury—based on a polynomial identity in k variables—obtained a closedform expression for the entries of the powers of a k-square matrix, usingthem to derive various combinatorial identities [16]; tractable explicit formsare, however, only available for low values of k ≥ 2. Our approach does notlend itself to any such development beyond the 2× 2 case presented.4 SummaryKnown results involving Catalan polynomials afford the formulation of atheorem by McLaughlin for arbitrary exponentiation of a general 2×2 ma-trix. The proof presented here—utilising the efficacy of the polynomials—isnew and differs significantly in style from McLaughlin’s (offering more in-sight into the result itself, too, we would suggest).References[1] Larcombe, P.J. (2015). A note on the invariance of the general 2 × 2matrix anti-diagonals ratio with increasing matrix power: four proofs,2Certainly, these polynomials—seen in papers by Webb and Parberry [9] and Bicknell[10]—are those for which Wikipedia and Wolfram MathWorld currently provide ‘hits’,and are themselves related to Morgan-Voyce polynomials (first introduced by that authorto analyse ladder networks using Fibonacci numbers in 1959); note that Hayes [11,Preface, p.ii] adopts a scaled version of the governing recursion which results in modifiedpolynomials that constitute a small variation within this class of modern day Fibonaccipolynomials. Koshy [8, Chapter 37] states that they were first studied by E.C. Catalan(as does Hayes [11]), being introduced as the first generalisation of the Fibonacci numbersto a set of polynomials. Benjamin and Quinn note that their coefficients—including zerosfrom terms not present—have a combinatorial interpretation in the context of tilings [12,Combinatorial Theorem 12, p.141].Fib. Quart., 53, pp.360-364.[2] McLaughlin, J. (2004). Combinatorial identities deriving from the n-th power of a 2 × 2 matrix, Int.: Elec. J. Comb. Num. Theory, 4,Art. No. A19, 15pp.[3] Larcombe, P.J. and Fennessey, E.J. (2015). A condition for anti-diagonals product invariance across powers of 2 × 2 matrix sets char-acterizing a particular class of polynomial families, Fib. Quart., 53,pp.175-179.[4] Clapperton, J.A., Larcombe, P.J. and Fennessey, E.J. (2012). Newclosed forms for Householder root finding functions and associated non-linear polynomial identities, Util. Math., 87, pp.131-150.[5] Clapperton, J.A., Larcombe, P.J. and Fennessey, E.J. (2008). On iter-ated generating functions for integer sequences, and Catalan polyno-mials, Util. Math., 77, pp.3-33.[6] Clapperton, J.A., Larcombe, P.J. and Fennessey, E.J. (2010). Newtheory and results from an algebraic application of Householder rootfinding schemes, Util. Math., 83, pp.3-36.[7] Jacobsthal, E. (1919/20). Fibonaccische polynome und kreisteilungs-gleichungen, Sitz. Berl. Math. Gesell., 17, pp.43-51.[8] Koshy, T. (2001). Fibonacci and Lucas numbers with applications,Wiley, New York, U.S.A.[9] Webb, W.A. and Parberry, E.A. (1969). Divisibility properties of Fi-bonacci polynomials, Fib. Quart., 7, pp.457-463.[10] Bicknell, M. (1970). A primer for the Fibonacci numbers: part vii (Anintroduction to Fibonacci polynomials and their divisibility proper-ties), Fib. Quart., 8, pp.407-420.[11] Hayes, R.A. (1965). Fibonacci and Lucas polynomials, M.Sc. Thesis,San Jose State College, U.S.A.[12] Benjamin, A.T. and Quinn, J.J. (2003). Proofs that really count: theart of combinatorial proof, Math. Ass. Amer. (Dolc. Math. Expos.No. 27), Washington, U.S.A.[13] Blatz, P. J. (1968). On the arbitrary power of an arbitrary (2 × 2)-matrix, Amer. Math. Month., 75, pp.57-58.[14] Williams, K.S. (1992). The nth power of a 2 × 2 matrix (in Notes),Math. Mag., 65, p.336.[15] Belbachir, H. and Bencherif, F. (2006). Linear recurrent sequencesand powers of a square matrix, Int.: Elec. J. Comb. Num. Theory, 6,Art. No. A12, 17pp.[16] McLaughlin, J. and Sury, B. (2005). Powers of a matrix and combina-torial identities, Int.: Elec. J. Comb. Num. Theory, 5, Art. No. A13,9pp.

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A new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power

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A new formulation of a result by McLaughlin for an arbitrary dimension 2 matrix power

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