THE BRAHMAGUPTA POLYNOMIALS IN TWO COMPLEX VARIABLES* In Commemoration of Brahmagupta's Fourteenth Centenary

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1 THE BRAHMAGUPTA POLYNOMIALS IN TWO COMPLEX VARIABLES* In Commemoration of Brahmagupta's Fourteenth Centenary E. R. Suryanarayan Department of Mathematics, University of Rhode Island, Kingston, EJ {Submitted April 1996-Final Revision March 1997) 1. INTRODUCTION Some of the properties of the Brahmagupta matrix [see (1) belo], and polynomials x n and y n in to real variables (x, y) (see 3) have been studied in [6]; e kno that the Brahmagupta polynomials contain the Fibonacci polynomials, the Pell and Pell-Lucas polynomials [2], [5], and the Morgan-Voyce polynomials [4], [7]. The convolution properties that hold for the Fibonacci polynomials and for the Pell and Pell-Lucas polynomials also hold for Brahmagupta polynomials. In this paper e extend analytically the properties of the Brahmagupta matrix and polynomials derived in [6] from to real variables to to complex variables and, hich belong to to distinct complex planes. We denote this space by C 2. A typical member in C 2 has the form = (,). Since C is simply R 2 ith the additional algebraic structure, e realie that C 2 is (topologically) R 4 ith some additional algebraic properties. We have a natural ay to identify points in C 2 ith points in R 4. This is described by the scheme: C 2 3(, ) <r+ (x+iy,u + iv) «-> (x, y, u, v) er 4 In particular, e measure the distance in C 2 in the customary Euclidean fashion: if C, x -( x, x ) and 2 - ( 2> i) a r e poi^s in C 2, then \g x = (\ \ ~ i \ \ ~ 2 2\l/2! ) Another interesting feature of the Brahmagupta polynomials n and n in C 2 is that, hen the polynomials are expressed in terms of real and imaginary parts ith - x + iy and = u + iv, the resulting polynomials x m y m u m v n satisfy recurrence relations (11)-(18). The functions x m y n, u n, v n are solutions of the second-order partial differential equations (19) and (20). Since the calculations go through ithout change in the complex case, e just list some of the properties. 2. BRAHMAGUPTA MATRIX Let B be a matrix (a Brahmagupta matrix) of the form B = \ t here t is the fixed real number and and are complex variables; further, e shall assume that deti? = p = 2 - t 2 * 0. Using its eigenrelations, B has the folloing diagonal form: t V2 \ 2 ^2" -"V2" Z + W«Jt 0 0-4t V2 V2f V2" ~V2F_ (1) * This paper, presented at the Seventh International Research Conference held in Gra, Austria, in July of 1996, as scheduled to appear in the Conference Proceedings. Hoever, due to refereeing problems and deadline dates, e are publishing it in this issue of The Fibonacci Quarterly to assure its timely publication. 34 [FEB.

2 Define B n = t Then the above diagonaliation enables us to compute Z n \t Z B Since B n+l B n = V2 V2 \+jt 0 V2 vit 0-4t\ - B n B, e have the folloing recurrence relations: +l = n + t n9 n+l = n + n, ith n - and n =. From (2) e derive the folloing Binet forms for n and n \ (2) (3) (4) W = 2 ^ [ ( Z + V 7 r " ( Z " V 7 r L (5) and n ± 4t n = ( ± 4t) n. Note that if e set = 1 / 2 = and t = 5 then /? = - 1 ; then 2 n = F n is the Fibonacci sequence, hile 2 n = L n is the Lucas sequence, here > 0. Let % = + 4i,rt = -4t,% n = n + n 4i, rt n = n - n 4i and fi n = 2 n-t 2 n, ith? = 7, t; n -, and /? = /?. Then e have % n = g n, r\ n = rf, and fi n = /?". To prove the last equality, consider/?» = ( 2 -fti/ 2 )» = ^ V = ^ I I = ( ^ - ^ 2 ) = A- We also have the folloing property: ob -. et + ei -j r (e 4 -e' } ) V?' Jtf^-e") e* + e* dete B =e 2. To prove these results, set l k = % k + if, 2*Jt k = % k -rj k. Since i=0 ' B k and B k 1 k\ k\ e express k and t in terms of and 77 to obtain the desired results. Recurrence relations (3) also imply that and satisfy the difference equations: twu * +i = 2* - P _ x, n+1 = 2 n - p n _ v (6) Conversely, if 0 = 1, x =, and 0 = 0, and x =, then the solutions of the difference equations (6) are given by the Binet forms (4) and (5). The expressions n and n can be extended to negative integers by defining _ n = n /3~ n and _ n = ~ n p~ n. Then e have t here e have used the property B~ n = t Z -r, t = B_ 1998] 35

3 t ~HY 1 - -t -t -M> All of the recurrence relations extend to the negative integers also. Notice that B = /, here / is the identity matrix. For negative integers, n and n are rational functions of and. 3. THE BRAHMAGUPTA POLYNOMIALS Using the Binet forms (4) and (5), e deduce some results: Write n and n as polynomials in and using the binomial expansion: + The first fe polynomials are 0 = 1, ^ =, 2 = 2 + T 2, 3 = 3 + 3f 2, 4 = 4 + 6r t 2 4,..., 0 = 0, ^, 2 = 2, 3 = f 3, 4 = t 3,... Notice that n and are homogeneous in and ; therefore, they are analytic (in the classical one-variable sense) in each variable separately. Also, n and n satisfy the Cauchy-Riemann equations in each variable separately: If n = x n + iy n, then and dx dy" dy dx ^ L = ^ L fan = fyn du dv ' #V ^ ' Similar relations are satisfied by the polynomials n = u n +iv n. If t > 0, then n and satisfy: lim -»oo V +VF if -V? if Z-yftW Z+yjtW < i >l; lim lim n-±co Z n -\ «-»co ^n-\ + >Jt if - V7 if -VF Z+y[tW +V^ <1, 6?Z?W n, n-l> a ~ t d~ ntw n-y From the above relations, e infer that and n are the polynomial solutions of the "ave equa tion": d 1 I a 1 \ d 1 t d 2 ) u = o. (7) 36 [FEB.

4 Since the partial differential equation (7) is linear, by the principle of superposition its general solution is 00 U(,) = ^(A rl +B n ), here 4, and B n are constants RECURRENCE RELATIONS From the Binet forms (4) and (5), e record the folloing obvious recurrence relations: ( 0 Z m+n= Z m Z n +tw m W n, 0 " ) m+ + P " m_ n = 2 W m, 09 W m +n = m W n + ^ m (VU) Z m+n + B n Z m _ = 2tW m W n, (iii) fi n m. n = ^n-t m v Ht - (viii) m+ + B n m _ = 2 m, ( 8 ) (iv) P" m _ = n m - m n, (ix) 2( 2 m- m+n m _ ) = p^^(p"- 2n ), (V) Z^+Z'V^V. 00 Z 2m~^m^m-n= P^^ln- Putting m = n in (i) and (ii) above, e see that 2 = % + t% and 2 = 2 n ; these relations imply that: (a) 2n is divisible by n ±iji n if 7>0; (b) 2 is divisible by n ±Ji n if *<0; (c) 2 is divisible by and and, if r divides s, then m and rn are divisors of sn. Let Ei=i = 2. Then, using the Binet forms, it is not difficult to see the folloing facts: (i\ y. -P n- Z n+1+ Z -P (m) 2 ^ - 2 ( ^ _ 2 Z 2 + 1) ) ' (iv) Zjt= ^ / <? 2 2f(/? l) 2*09-1)' (v) 2Z 1 k +l _ k =n +l +i B (vi) 27 Z WfcW^i-it = "2 n +i - - * W (vii) 2 2 ^ _^+1 = 2 Z iv* v*+i = rm ] n+v Mo e generalie a result satisfied by the generating functions of Fibonacci (F n ) and Lucas (L n ) sequences; namely, i n Then L(f) = e 2F^ [3]. A similar result holds beteen n and n. Let Z and JF be generating functions of n and n, respectively; that is, i 1998] 37

5 Z =ST S "' ^=2>x- (9) Then W(s) = se 2Z^. Since the proof is similar to the real case (see [6])? e omit it here. 5. SERIES SUMMATION INVOLVING RECIPROCALS OF n AND n All the properties of infinite series summation involving x n and y n can be extended to the complex variables case also. Since the arithmetic goes through ithout any changes, e shall just list them here. For details, see [6]. 1. k=l k=r+l 1 ( 2 jg + l = 1 k J ' 2 Zv,Zi \^k-l^k+l 1 ^ 7 = 1 k=r+l Z k-l Z k+l k=r+l \ k-l CO -j k = 2 2 r Z ( t + l ) r ^ Z(fc-l)r I±L\ -_L Zh±\Zh ^k+l^k r r^r+1 > Z ' Z k+l) 2 P±L\- ktr + \\ W k-l W k+\ Wk+lWkJ W r W r+l ± + J- f 2 k = j J_+^+l ±I V_i_ Jfcr J Z r Z 2r oo 1 i - l kll 2 r W,. W (k+\)r y W (k-l)r J3 r + l \, kr l 1 W r W. r n lr 6. ri. o2*-'-2 1 fc=2 J 2 * (x+yjt) 2 ' I i «=1 Z n Z n+k = 7r-fi 5l:=L -*(^±^)\ here the plus sign should be taken if \%lrj\< 1 and the minus sign should be taken if \%lrj\> 1. To sho item 6, e consider Thus, Z n-l Z n+k ~~ Z n+k-l Z n ~ y l" _ 1 _,, n-rn+k [ n n+k tw k fww. /WW. "*-* \ Z n-\ { +k _ x + t n+k _ x ) - ^., ( : v i + h j Z r Z n Z n + k ^n+k-fn t h ( k N+k ILT-l^ N o fixi>l and let Ntend to infinity. Using the property e derived in Section 3, e obtain the required result. Similarly, e sho that here the plus sign should be taken if / rj \ < 1 and the minus sign should be taken if / 771 > [FEB.

6 6* CONVOLUTIONS FOR n AND n Suppose that a n (, ) and b n (, ) are to homogeneous polynomial sequences In to variables and, here n is an integer > 1. Their first convolution sequence is defined by (fln * b J l) n = S a j b n+l-j = S h a n+l-j ' In the above definition, e have ritten a n = a n (, ) and b n = b (, ). To determine the convolutions n * n, n * n, and n * n, e use the matrix properties ofb, namely, Let t -1/f+l n+l W n+i :B n+l =B J B n+l - J ' = n Zj twj Wj Zj n+l-j twn+l-j n+l d) a) 1=1 J=l t 2 (1). «n Note that B" - B n. We prefer using the subscript notation. Since H" J=1 B n+1 =nb n+l, the above result can be ritten as W n+l-j Z n+\-j n B + i = n* r,+ iw n* W n 2t * n B?\ here e have ritten Z"=i = 2. Therefore, e have jp = n n+l and jp = n n+h or 2 * =m +l +^ and 2t n * n =m n+l -*~^, from (8) parts (v) and (vi). The above result can be extended to the k^ convolution by defining We can sho that 7=1 tf^f'tv We shall prove the result by induction on fc. Since Z? (1) = nb n+l, the result is true for k = 1. No consider ^ +, ) = Z 5 y ^ = Z ^ ( ^ ) ) = 2 $ '«+i- ^y'+fc ~ ^i+jfc+1 Z y + Ar-lV f«+ * it k+ir n+k+l hich completes the induction. We have used the property Zf 7 "^-1 ) = ( +*), to derive the above result. From the above results, e can rite the folloing k^ convolutions, namely, (k) (n + k-l) i a) (n + k-l) «I k p+* and» { k J»+k- (10) 1998] 39

7 Result (10) enables us to rite the convolutions n *^k\ n^^\ e shall sho that n^^h\ and n * ^k\ First, We consider 2 *«=2Z* _, +1 \ Z j+k Z n-j+l -2 f c S * - ^ «(j + k-l y _ J ^ X, y=i V * -2j+l + ^ 2 ^ + A-/+i 1 l(h' B r t -^f_ 2 J + 1 ) =?,[ j + k k~ ^ ( V ^ i + t^xfi* {j + l~ i y k _ 2j+l -t k _ 2j+l ) = \jfc + lj Z "+*+l +^ k )P J+ Z n+l-2j-k- We have used (10) and (8) part (i) to derive the above result. Similarly, e can sho that n+l-2j-k> n+\-2j-k' 7. THE IMPLICATIONS OF n AND n IN R 4 Let = x+iy and = u + iv. Then = x +i>, = W +JV W, and J3 = 2 -t 2 = a + iy, here a = x 2 - y 2 - t(u 2 - v 2 ) and y = 2(xy - tav). Note that det B * 0 implies that either a * 0 or y * 0 Recurrence relations (3) no become: **+l = 2 ** " 2 m - «V l +?%-!, 0 1) ^+1 = 2 ^ % - ^.! - ^.!, (12) + i = 2xu n - 2yv n - au n _ x + yv n _ h (13) V i = 2*v + 2j/ - yu n _ x ~ a Vi> ( 14 ) 40 [FEB.

8 ith x 0 = 1, y Q = 0, u 0 = 0, v 0 = 0 and x x = x, y x =y, u x =u, v x = v. By (11)-(14), the first fe polynomials are given by x 2 = x 2 - y 2 + t(u 2 - v 2 ), y 2 = 2(xy + tuv\ u 2 =2(xu-yv\ v 2 = 2{xv+yu). x 3 = x 3-3xy 2 + 3txu 2-3txv 2-6tyuv, y 3 = 3x 2 y-y 3 + 6txuv+3tyu 2-3tyv 2, 1% = 3x 2 u - 3y 2 u - 6xyv - 3tuv 2 + tu 3, v 3 = 6xyu + 3x 2 v- 3y 2 v + 3tu 2 v-tv 3,... By expressing equations (8) parts (i) and (ii) in terms of the real and imaginary components, e find that the recurrence relations transform to X «Hrn = X n X n ~ yjn +<*m V» " y m+ n = X rjn + ^ m + KWn "Wml U m+n = X m U n + X n U m ~ J V * " 3W»> U n V m\ (15) To transform the partial differential equation (7) in and to the one in variables x, y, u, and v, e use the partial differential operators: d_ d Then equation (7) becomes 2{dx l dy) d 2 d 2 \(d 2 d 2 dx 2 dy 2 t\du 2 du 2 and ' jl d = l(a+j d 2\dx dy fn=<>, (16) (17) (18) (19) r a 2 id 2 dxdy t duch g n = o. (20) here f n = x n or u n and g n - y n or v n. By the principle of superposition, the solution of differential equations (19) and (20) are, respectively, OO f(x,y,u,v) = ]T(a n x n +h n u n ) and g(x,y,u,v)=y*(c n y n +d v n ), 0 0 here a n9 b n, c n9 and d n are constants. No e express relation (9) in Section 4, i.e., W(s) = se 2Z^s\ in terms of real and imaginary parts. Let Z(s) = X(s) + iy(s) and W(s) = U(s)+iV(s). Then (9) transforms to and U(s) = use X{s \u cosy(s) - v sin Y(s)) V(s) = vse X(s) (v cosy(s) + u sin Y(s)). OO 1998] 41

9 No, let us turn our attention to the convolutions in Section 6. Result (11), expressed in terms of real and imaginary components, becomes x(k)_(n + k-l] u(k)-(n + k-l) ik)_(n + k--l\ (k)_(n + k-i) We have seen here some of the properties of the matrix B ith complex entries; e are sure there are many more of them. REFERENCES 1. R. C. Entringer & P. J. Slater. "Gossips and Telegraphs." J. Franklin Institute (1979): A. F. Horadam & J. M. Mahon. "Convolutions for Pell Polynomials." In Fibonacci Numbers and Their Applications 1: Dordrecht: D. Reidel, R. Honsberger. Mathematical Gems, pp Ne York: The Mathematical Association of America, J. Lahr. "Fibonacci and Lucas Numbers and the Morgan-Voyce Polynomials in Ladder Netorks and in Electric Line Theory." In Fibonacci Numbers and Their Applications 1. Dordrecht: D. Reidel, J. M. Mahon & A. F. Horadam. "Infinite Series Summation Involving Reciprocals of Pell Polynomials." In Fibonacci Numbers and Their Applications 1. Dordrecht: D. Reidel, E. R. Suryanarayan. "The Brahmagupta Polynomials." The Fibonacci Quarterly 34.1 (1996): M. N. S. Samy. "Further Properties of the Morgan-Voyce Polynomials." The Fibonacci Quarterly 6.2 (1968): AMS Classification Numbers: 01A32, 11B37, 11B39 42 [FEB.