THE BRAHMAGUPTA POLYNOMIALS. E. R. Swryanarayan Department of Mathematics, University of Rhode Island, Kingston, RI02881 (Submitted May 1994)

Transcription

1 E. R. Swryanarayan Department of Mathematics, University of Rhode Island, Kingston, RI0288 (Submitted May 994). INTRODUCTION In this paper we define the Brahmagupta matrix [see (), below] and show that it generates a class of homogeneous polynomials x n and y n in x and y satisfying a host of relations; the polynomials contain as special cases the well-known Fibonacci, Lucas, and Pell sequences, and the sequences observed by Entringer and Slater [], while they were investigating the problem of information dissemination through telegraphs; x n and y n also include the Fibonacci polynomials, the Pell and Pell-Lucas polynomials [5], [7], and the Morgan-Voyce polynomials in Ladder Networks and in Electric Line Theory [6], [9]. We also extend some series and convolution properties that hold for the Fibonacci and Lucas sequences and the Pell and Pell-Lucas polynomials to x n andj [7], [5]. 2. THE BRAHMAGUPTA MATRIX To solve the indeterminate equation x 2 = ty 2 ± m in integers, where t is square free, the Indian astronomer and mathematician Brahmagupta (ca. 598) gave an iterative method of deriving new solutions from the known ones by his samasa-bhavana, the principle of composition: If a r e (x^y^m^ and (x 2,y 2,^h) trial solutions of the indeterminate equation, then the triple ls (x l x 2 ±ty ] y 2,x l y 2 ±y l x 2,m l m 2 ) a ' s o a solution of the indeterminate equation which can be expressed using the multiplication rule for a 2 by 2 matrix. Notice that if we set and m = det B, then the results B(x,y) = \ x ±ty y ±x B(x u y l )B{x 2,y 2 ) = B(x lx2 ± tyy 2, xy 2 ±y x 2 ), det[b(x u y l )B(x 2, y 2 )] = detb(x u yjdet B(x 2, y 2 ) = m^, give the Brahmagupta rule. Equation (3) is usually referred to as the Brahmagupta Identity and appears often in the history of number theory [0]. Let M denote the set of matrices of the form x ty where t is a fixed real number and x and y are variables. Define B to be the Brahmagupta matrix. M satisfies the following properties:. M is a field for x, y,t GR and t < 0; in particular, if t = -, then we have the well-known one-to-one correspondence between the set of matrices and the complex numbers x + iy\ x y -y x <r^x+iy. () (2) (3) (4) 30 [FEB.

2 THE BRAHM AGUPTA POLYNOMIALS 2. The following eigenrelations, in which J denotes the transpose, hold: B[l,±Ji] T = (x±y-jt)[\,±ji) T, and these relations imply B"[l,±4t] T =(x±y4iyll,±^] T aind x y 42 V2 x+y4t 0 V2 yit ty x yl ~V"2. 0 x-jv? yl \2F. Define B n x y = :B n. ty x tyn X n. 3. Then the following recurrence relations are satisfied: *A+I = ^n +t yy n > y n+ i = % + y* n > with x n - x and y n = y. 4. Using the above eigenrelations, we derive the following Binet forms for x n and>> : x n = h(x+yjly + (x-y^yl y»=^ji( x+ y^y~( x -y^yi (5) (6) (7) andx n ±Jty n = (x±jiy) n. 5. Let % n = x n +y n <Ji, r\ n -yji, and P n = x 2 n-ty 2 n, with ^ = 77,, = and /? = /?; we then have 4 = <f, 77 = if, n =@ n. To show the last equality, consider /?" = (x 2-0/ 2 )" = V = Zfln = ( x l ~ Od = P n - Notice that fi = det5. 6. The recurrence relations (5) also imply that x n and y n satisfy the difference equations: x w+ = 2xx n -fix^l9 y n+ = 2x j -fiy n _ v (8) Conversely, if x 0 =, x i = x, and j 0 = 0, y x -y 9 then the solutions of the difference equations (8) are indeed given by the Binet forms (6) and (7). 7. Notice that if we set x = /2 = j a n d / = 5, then fi = -l and 2y n -F n is the Fibonacci sequence, while 2x n = L n is the Lucas sequence, where «>0. For the number-theoretic properties of F andz,, the reader is referred to [2] and [3]. 8. In particular, if x = y = and t = 2, then both x andy n satisfy x 2-2 j 2 = (-l) n and they generate the Pell sequences: x n =,,3,7,7,4,99,239,577,..., y n = 0,,2,5,2,29,70,69,408,... It is interesting to note that if we set a = 2(x n +y n )y n b = x n (x n +2y n \ c = x 2 n+2x rl y n +2yl 996] 3

3 we then obtain integral solutions of the Pythagorean relation a 2 +b 2 =c 2, where a and h are consecutive integers [8]. 9. If t =, then x n +y n = (x+y) n 9 and if t = -, then x n +iy n = (x + iy) n. Also, for every square free integer t, the set of matrices M is isomorphic to the set {x+yjt\x,y G Z }, where Z is the set of integers. 0. 4» j+ - ov e' 7 ^ ( e * - e ' ), dete" = e^. To show these results, let us write 2x fc = ^ + 77*, 2^0^ = ^ - if Since CO Z>fc A! and ^ = J_ * = 0 ** yk tyk x k. we express x k andj^. in terms of and 7 and obtain the desired results.. x andy can be extended to negative integers by defining x_ n = x n B~ n and >*_ = -y ft~". We will then have 2T here we have used the property (r --Y x y ty x J x ty y x x -y -ty *- y- n x All the recurrence relations extend to the negative integers, also. Notice that B -, the identity matrix. TT P" B x» -y» 3. THE BRAHMAGUPTA POLYNOMIALS. Using the Binet forms (6) and (7), we can deduce a number of results. Write x n and y n as polynomials in x and y using the binomial expansion: Xn=x» + tfyx»-y + t 2 (>i)x»-y + ---, y = m"- l y + t(^jx"-y + t 2^y-V + - Notice that x n and y n are homogeneous in x andj. The first few polynomials are x 0 =, x x = x, x 2 - x 2 +ty 2, x 3 - x 3 + 3tyx 2, x 4 - x 4 + 6tx 2 y 2 +t 2 y 4,..., Jo = > y\ = y, yi = 2x y, ys = ix 2 y+*y 3, y 4 = 4x 3 y+4txy 3, If t > 0, then x and y n satisfy l i m ^ V f, lim-^-=lim J«: + VFy. ^»» - > «^ i "-**.y«-i 32 [FEB.

4 3. dx dy : «V i, ay 'dy " ^ From the above relations, we infer that x n and>> are the polynomial solutions of the wave equation d 2 i d 2 ^ u = o. ^dx l t dy 4. If/? = -, then ty 2 = x 2 +, then the difference equations (8) become X n+l= 2xX n+ X n-l> JVHI = 2 % + JVl ( 9 ) 5. If 2x = a and /? =, x x =, x 2 = a, and y x =, j 2 = a -, then x and j / w generate Morgan- Voyce polynomials [6], [9]. 4. RECURRENCE RELATIONS. From the Binet forms (6) and (7), we can derive the following recurrence relations: 0) x m n = x m x n + ty m y n9 (iii) ^ X - K ^ V K - O ^, (iv) ^ " ^ - ^ x ^ - x ^, ( v ) *m+w ~*~ P X m-n ~ ^Xm X m (Vi) yn*n+p n yi*-n= 2x nym> (vii) x w+ - /?%,_ = 2 0vy*> (io) (viii) J^"^> m _ w =2x w j;, (ix) 2(x 2 m - x m+n x m _ ) = p m ~ n U3 n - x 2 J, (x) x 2m - 2ty m+n y m _ n = ^w~"x 2w. 2. Put m = n in (i) and (ii) above; then we see that and these relationships imply that (a) x 2 is divisible by x n ±i4iy n, if t > 0, (b) x 2n is divisible by x n ±i<jty n9 if t < 0, (c) j> 2rt is divisible by x and y n \ also, if r divides s, then x r and^r are divisors of y sn. 3. Let Sjt=i = E. Then, using the Binet forms, we can also derive the following relations: 996] 33

5 W 2*y k - p_ 2x+l, (Hi) ^x2 ^n-^^-f +Et^_ V ' k 2 ( ) J C 2 + ) 2(y?-l) ' K } ^ y k 2f(/? 2-2* 2 +l) 2t(fi-l)> (v) 2l,x k x n + x _ k =nx n+l +^, (vi) 2f I ^ ^. * = m: + - ^, (vii) 22x^w_^+ - 2Z ^ V * + i = W JW en) 4. Now we show an interesting result which generalizes a property that holds between F n and L n, namely, e L^ = F(x), where and F(x) = F l +F 2 x + F 3 x F +l x" + -, L(x) = Lx + ^x 2 +^-x 2 w l + +^-x" n (see [4]). Let X and 7 be generating functions of x n and_y, respectively; that is, n then Y(s) = sye 2X^. To prove this result, consider Y(s) = y x s+y 2 s 2 + y 3 s* + + y^sp +. Then sy(s) = y^+y 2 s?+y 3 s y n $" +l + --, md ^(s) = y/ + y 2 $ y^* 2 +. Substituting the power series for Y(s) into the expression Y(x)- 2xsY(s)+fls 2 Y(s), we obtain [l-2x5+^]7(5) = j5 + [ j, % + ^ _ ] / +, where we have put y x =y. Now, using the property y k+l - 2xy k + fty k _i = 0 in equation (8), we find that the above expression reduces to Now consider the series [l-2xs+j3s 2 ]Y(s) = ys. (2) w i 2 3 and in it express x n in terms of % n and rj n to get n X(s) = ±( + 77)5+ I ^ + ^ V + i r i ^ + r/ 3 ) ^ n which can be rewritten in the form ^ n + rf) *" [FEB.

6 THE BRAHM AGUPTA POLYNOMIALS X(s) = / V Therefore, X(s) = - l n ( l - ^ Since ( - s)(l-r/s) = l-2xs+/fc- 2, we have 2X(s) = -ln[l-2xs + /fe 2 ]. (3) Now compare (2) and (3) and obtain the desired result: Y(s) = sye 2X^. 5. SERIES SUMMATION INVOLVING RECIPROCALS OF x w AND y n Let us look at some infinite series summations involving x n andj and extend some of the infinite series results that are known for F n and L n [4] and for P (x) an d Q n ( x ) Ul to x n a n dj. First, we shall show that. 00 -J X k=l X k+l To show the above result, consider 2x /? +!] _ V X A r - l *-k+l *k-l X k-l X k X k X k+l X k-l X k X k+l _ ^xx k ~ P*k- ~ X k-l _ 2X X 'k-l k X k k+l X X k-l X k+l fi+i Xj r Xi ^k^k+l where we have used the property x k+l = 2xx k - Px k _ x. Therefore, ^ jfc=l X k+ \ X k-l ( 2x fi + l\_^( jfc=lv * * -! * * x k x k+lj x ii x l * In particular, if ty 2 = + x 2 and j =, then p = -, and the above result reduces to k=i Xu_\Xu ^k-n+i 2x 2 ' where x fc is given by equation (9). Similarly, we can show that 2. 2x fi + l ) k=r+l\ X k-l X k+l X k+l X kj X r X r+l For the special case ty 2 = + x 2 and y = l, then /? = - and the above result becomes y i = k=r+l X k-l X k+l 2 x X r X r + where x k is given by equation (9). Following a similar argument, we can show 996] 35

7 THE BRAHM AGUPTA POLYNOMIALS 3. 2x /? + l Again using 2xx k = x k+l +Px k _ x, we can derive y \ X k-l X k+l If fy 2 = + x 2 and _y =, then we have 3 L 2xx k _ y f i- +^- k=r+l Xlr-lXlr- x k-\ x k+l 2x \ X r + where x k is given by equation (9). Similarly, from the recurrence relation 2xy k - y k+l +ffy k i, we have 2xy k i- + -A k=r+i yk-iyk+i j t = r + i v ^ - i yk+i j In particular, if ty 2 = + x 2 mdy =, then /? = - and the above result becomes X ' r+ k=r+l where y k is now given by equation (9). 6. Now we generalize the results of items 2 and 3 of this section; we shall show that IT Ar=2 X (k+i)r 2x r /r+i \ X (k-l)r K kr To show this, we consider the left-hand side of the above result: k=2 X (k+l)r y I 2x r X kr ~ P X {k-\)r ~ X (k-l)r X (k-l)r X kr J The above result can be simplified by using property (v) of (0), with m = rfc, n = r, that is, x rk+r + P Tx rk-r = ^Xrk x r- Then the above expression becomes co i IT (~ -r ^ (k+l)r x {k-\)r k=2 X (k+l)r V X (k-l)r X kr J which reduces to k=2\ X (k-l)r X kr X kr X (k+l)r J which, when summed over k, reduces to / (x r x 2r ). Similarly, we show that oo i k=2 y(k+l)r y(k-l)r yjcr y r y 2r 36 [FEB.

8 7. Let us now generalize a property that holds for Fibonacci series [4]. For t > 0, consider the series: Denote s=ffi*' l - 2 y 2= y 2 { /S 2 y 2 ^y 2 h. y 2 * y 4 y% y u By induction we shall show that y* y% J i 6 y 2 n V - ^2"-2 5 ""V Note that j 2 = 0 implies that either x = 0 or y - 0 from the Binet form for j 2. Therefore, we shall assume that y T ^ 0. Observe also that equation (2) is true for n = 2, 3. Consider (4) y* y 7 n y~ t y»y, 2 nj 2 n Use the property y 2m = 2x m y m, with m = 2 n+l = 2(2"), to get $n+l ~ _ 2x ry r y r _ 2 + y r fi y 2 _ 2x ry r _ 2 + P y 2 y^y, 2nj2n Now recall property (viii) of equation (0), y m+p -j3 p y m _ p =2x m y p, and in it set m-2 n, p = 2 n -2. We then have _ 3 ; 2«+i_ 2 J 2 n+ which completes the induction. Therefore, for t > 0, we have y on+ S= lim = lim ^ - = ^-2 = ^. _>«> ->«> v (X + JVO 6. CONVOLUTIONS FOR x n AND y n Given two homogeneous polynomial sequences a n (x,y) and b n (x,y) in two variables x and j>, where n is an integer >, their first convolution sequence is defined by n (a n *bj l) = Y, a fin+i-j = X * A + w In the above definition, we have written a n =a n (x 9 y) and ft w = h n (x, y). Denote x n *x n - Xfp, y n^y n - Y^l\ 2x n *y n =y \ and Xjp +ty^l) = x }. To determine these convolutions, we use the matrix properties of B, namely, n (y x.-iw+ r X «+l.vw+ " I = 5^ = 5/5^-/= I X n+-j tyn+l-j y^+l-j X n+l-j 996] 37

9 Let y=i 7= Note that B" = B n. We prefer using the subscript notation. Since " = 2? n+ = ^ +? we have 77+ X n+l-j ; = i (Vy Xj Jyn+l-j Let Z" = = S, then the above result can be written yn+l-j X n+l-j_»s + = [L Xj-y +l _j + Z, yjx n+l _j J z, XjX _ j+tzyjy n+l -j or «4,+i =. 2tx *y x n *x +ty *y ^ Therefore, we have *J } =**«+!> yn )=n y^v ^ ty The above result can be extended to the k^ convolution by defining yd) x = B?\ Now we shall show that 7= w-ri- ]^. We shall prove the result by induction on k. Since J5 () = nb n+u the result is true for k =. Now consider =^»w( j + r>;«=**.^+**">(i;^ w+fc+? which completes the induction. From the above results, we write the k^ convolution of x n and^: >.(*). n + k-l k \ x n+k> y n - Also, from properties (v) and (vi) of (0), we have,(k)_[n+k-r 2X () = nx + + ML &*-, 2tt- l) y = nx +l - y ' which can be written in the form (5) (6) 2XU = Py n Py XW+^, 2 ^ = ^ - n 38 [FEB.

10 We can also extend the above result to the k^ convolution of x n and j, namely, Using the results (0)(v), (5), and (6), and some computation, we obtain Similarly, we have X n+l-2j-k' LX yn*yn ~\ k + l) X "+k+l M Jc \P X n+l-2j-k> 2xW*v ~( n + k + )v +y(j k ~ l ]fr +k v LX n yn~[k + l) S"+k+l + 2 \ k )" yn+l-2j-k > 2x*v^-( n + k )v -y(j + k ~ l )BJ +k v What we have seen here is but a sample of the properties displayed by the versatile matrix B. We are sure there are many more. REFERENCES. R. C. Entringer & P. J. Slater. "Gossips and Telegraphs." J. Franklin Institute 307,6 (979): G. H. Hardy and E. M. Wright. An Introduction to Number Theory. Oxford: Oxford University Press, V. E. Hoggatt, Jr. Fibonacci and Lucas Numbers. Boston: Houghton Mifflin, R. Honsberger. Mathematical Gems, pp New York: The Mathematical Association of America, A. F. Horadam & J. M. Mahon. "Convolutions for Pell Polynomials." In Fibonacci Numbers and Their Applications. Dordrecht: D. Reidel, J. Lahr. "Fibonacci and Lucas Numbers and the Morgan-Voyce Polynomials in Ladder Networks and in Electric Line Theory." In Fibonacci Numbers and Their Applications. Dordrecht: D. Reidel, J. M. Mahon & A. F. Horadam. "Infinite Series Summation Involving Reciprocals of Pell Polynomials." In Fibonacci Numbers and Their Applications. Dordrecht: D. Reidel, C. D. Olds. Continued Fractions, p. 8. New York: Random House and the L. W. Singer Company, M. N. S. Swamy. "Further Properties of Morgan-Voyce Polynomials." The Fibonacci Quarterly 6.2 (968): A. Weil. Number Theory: An Approach through History from Hammurapi to Legendre. Boston: Birkhauser, 983. AMS Classification Numbers: '0A32, B37, B39 996] 39