REPRESENTATION GRIDS FOR CERTAIN MORGAN-VOYCE NUMBERS A. F. Horadam The University of New England, Annidale, 2351, Australia (Submitted February 1998)

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1 REPRESENTATION GRIDS FOR CERTAIN MORGAN-VOYCE NUMBERS A. F. Horadam The University of New Engl, Annidale, 2351, Australia (Submitted February 1998) 1. BACKGROUND Properties of representation number sequences {2S }, {%,} associated with the Morgan- Voyce polynomials B n (x) the related polynomials C n (x) were recently investigated in [1]. Hopefully, the notation references in [1] will be accessible to the reader. Complementary properties of the number sequences {b n }, {c n } associated with the Morgan- Voyce polynomials h n (x) the related polynomials c n {x) are now explored. With x = 1 in these just-mentioned polynomials, we define the resulting numbers by Accordingly, these numbers are K^K-x-K-i, h = \,b x = \, (i.i) c «= 3 ^ r V 2» % = -\ <i = l. (1-2) H = , b n = , (1.3) c n = Consider now the unit coefficient representation sums for b n, c n analogous to those for B n, C n [1]. Irrespective of the uniqueness or otherwise of the representations ( of questions of minimality or maximality), we may assert that, for the representation number sequences {fo }, K = t J K=F 2n = F n L n (1.4) ^ Xy M n uuu, in terms of the Fibonacci Lucas numbers F, L n. Elements of {b }, {c } are thus n = 0 b = 0 c = 0 i, = Z ^ = ^. - 2 = i " (1-5) /=i , Why, we may ask, are these numbers worthy of our consideration? Firstly, as mathematical constructs they have an inherent interest to the inquiring mind ("because they are there"!). Secondly, as the theory necessarily compact unfolds, they add a little, however modest, to our knowledge of number relationships. Moreover, they complete the theme initiated in [1]. (1.6) 320 [NOV.

2 2. PROPERTIES OF h, c n W? ft One may readily establish the fundamental infrastructure of these two number systems, details of which are herewith reported (in pairs, for comparison). Recurrences: Binet forms: K=3K-i-^n-2, (2.1 In _ c n = 3c^1-c l _ (2.2 oln K= a_ P p {ap = \ap=-l\ (2.3 Generating functions: c n = a 2n +j3 2n -2. (2.4 J^b^-^ll-iSx-x 2 )]- 1, (2.5 Simson formulas: Summations: f ] c^7" 1 = (1 + x)[l - (4x - 4x 2 + X 3 )]" 1. (2.6; b ^ i b ^ - b ^ - 1, (2.7; c + ic -i-c^ = l-2c. (2.8 I b ( = F M - l, (2.9; I c, = Z (2«+ l), (2.10; 7=1 i b 2, = ( i ), (2.11 7=1 I c 2 / = F 4n+2 -(2«+ l), (2.12; 2 X - i = ^, (2.13 Zc 2,_ 1 = F 4-2», (2.14; i(-tf + %=jv-(-iri*»il (2.i5 I(-l) / + 1 c,=(-l)"[l-f ]. (2.16; 1999] 321

3 Other simple properties: K C- = » -K-i c - c - i b «+ b «+l = «n + c - i = 5^2n-l " 4, K <v b - 2 ~ C - 2 = F 2n - = L 2 - : ^ 2 w - l -1? = ^ - = 5F 2. 1> also, -2? -2? (2.17) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) (2.24) Next, we introduce the concepts ^ - ^ 1 = ^4-2 = ^ 2-1 ^ 1 - (2-25) 3. THE REPRESENTATION GRIDS b n = b» + l + b - l = 3b = 3F 2 = w n - > _ l, (3.1) (3.2) on invoking [1]. Repeating the summation process developed in [1], i.e., bj,' = h f n+l +b f rl_ l, we eventually arrive at the more general notations b<"> = b > 1+ b<3 (bf=b ), (3.3) ci ffl) = «+ cw (<f = c ). (3.4) As in [1], these data can be organized in (representation) grids for b ^ c^\ where m denotes columns n rows. Various approaches allow us to validate the properties recorded below, some of which are readily obtainable from the patterns in the rectangular grids, which the reader should construct for visual emphasis clarification of the theory. Zero subscripts: Negative subscripts: bw = 0 = S8 ( 0 0), (3.5) c(m) = 2(3 W -2 ) = 288 ( 0 7w) = -2%^ by [1]. (3.6) b ^ = - b l m ), (3.7) c? = c»>. (3.8) 322 [Nov.

4 Recurrences: (columns) fbs">=3bs3-b&, {cw = 3cW-cW+2'" +1, b = 3»'b n = 3'»F 2 <=1h<T\ (rows) \ ci m) = 3 m c + cl m). Binetforms: b {^ = y{a 2n -p 2n )l{a-p), c ( ) = 3 m (a 2 "+J3 2 ")-2 m+l. Generating functions: bf "V - 1 = 3 m [l - (3x - x 2 )]- 1, ;=1 Simson formulas: CWJC'-I = 3»(3 _ 2JC)[1 - (3x - x 2 )]" 1-2 m+1. ;=1 bl: ),b(- ),-(b ( w) ) 2 =-3 2m ) c & a - l c ^ ) 2 = 3 m {3 m -5-2^L 2n ). Summations: Ib<")=3»(F 2 B r t -l), 1=1 O^fter simple properties: Xc,( m) = 3 m (Z l)-2'" +1». 1=1 b ^ + b ^ - Z ^ also, c «+ c«=3 m -5F 2 _ 1-2(3'" + 2'"), cw_ b (m) = 2 aw. 4. FOREGROUND 1. Augmented Sequence Let us now recall, as in [1], the augmented sequence {2S* (a, 6, k) = 2S*} defined by 2T +2 (a, 6, A) = 3SC +1 (a, *, *) - % (a, *,*) + *. (4.1) 1999] 323

5 Initially, assume Hence, while *(a,b,k) = a, * 2 (a,b,k) = b. (4.2) 2C +1 (l,3,0) = b, (4.3) 8S; +1 (l,5,2) = c, (4.4) >* n+l (l,2,0) = b n, (4.5) a; +1 (l,4,0) = c. (4.6) 2. Brahmagupta Polynomials Very recently, Suryanarayan showed in [4] [5] how, by means of the Brahmagupta matrix, to generate polynomials x n y n (Brahmagupta polynomials) which include inter alia Fibonacci, Pell, Pell-Lucas polynomials, as well as the Morgan-Voyce polynomials B n (x) - x n b n (x) - y n described in [1] [2]. Suppose we express the vital difference equations [4, eqn. (8)], [5, eqn. (9)] in a slightly varied notation as x n + i = Px - Q v i, y n+i = Py n - Qy n -i (4.7) Selecting P = x + 2, Q = 1, x x -2, x 2 = P, y x = - 1, y 2 -\ (so y 3 = x + 3) in (4.7), we readily come to the polynomials C n (x) = x n c n (x) = y n, which [1], [2] are adjunct to B n (x) b n (x). 3. Further Developments These might profitably include, for instance, a) properties of b_ m c_ n (n > 0), b) extension of the theory to polynomials b (x), c (x) ( also ^ (x), ^^(x) [1]), c) construction of a representation table of sufficient scope to afford numerical enhancement of the patterns contained therein, d) uniqueness or otherwise of the representation, e) any additional Brahmagupta properties. 4. Associated Legendre Polynomials The author has become aware that the Morgan-Voyce polynomials b n (x) defined in (1.1) are essentially the associated Legendre polynomials p n (x) described by Riordan [3, p. 85]. In fact, K+i( x )~Pn( x )'-> e 8> b 3 (x) = p 2 (x) = l + 3x + x 2. Properties of p n (x) listed in [3] may then be cast in the b n (x) notation. Essential links for the equality of p n (x) b n+l (x) are the closed forms Chebyshev polynomials results in [3, p. 85] [2, (2.21) (4.14)]. 324 [NOV.

6 REFERENCES 1. A. F. Horadam. "Unit Coefficient Sums for Certain Morgan-Voyce Numbers." Notes on Number Theory Discrete Mathematics 3.3 (1997): A. F. Horadam. "New Aspects of Morgan-Voyce Polynomials." In Applications of Fibonacci Numbers 7: Ed. G. E. Bergum et al. Dordrecht: Kluwer, J. Kiordan. Combinatorial Identities. New York: Wiley, E. R. Suryanarayan. "The Brahmagupta Polynomials." The Fibonacci Quarterly 34.1 (1996): E. R. Suryanarayan. "Properties of the Brahmagupta Matrix." Int. J. Math Educ. Set Technol 27.3 (1996): AMS Classification Number: 11B37 Author Title Index Tile AUTHOR, TITLE, KEY-WORD, ELEMENTARY PROBLEMS, ADVANCED PROBLEMS indices for the first 30 volumes of The Fibonacci Quarterly have been completed by Dr. Charles K. Cook. Publication of the completed indices is on a 3.5-inch, high density disk. The price for a copyrighted version of the disk will be $40.00 plus postage for non-subscribers, while subscribers to The Fibonacci Quarterly need only pay $20.00 plus postage. For additional information, or to order a disk copy of the indices, write to: PROFESSOR CHARLES K. COOK DEPARTMENT OF MATHEMATICS UNIVERSITY OF SOUTH CAROLINA AT SUMTER 1 LOUISE CIRCLE SUMTER, SC The indices have been compiled using WORDPERFECT. Should you wish to order a copy of the indices for another wordprocessor or for a non-compatible IBM machine, please explain your situation to Dr. Cook when you place your order he will try to accommodate you. DO NOT SEND PAYMENT WITH YOUR ORDER. You will be billed for the indices postage by Dr. Cook when he sends you the disk. A star is used in the indices to indicate unsolved problems. Furthermore, Dr. Cook is working on a SUBJECT index will also be classifying all articles by use of the AMS Classification Scheme. Those who purchase the indices will be given one free update of all indices when the SUBJECT index the AMS Classification of all articles published in The Fibonacci Quarterly are completed. 1999] 325