DIFFERENTIAL PROPERTIES OF A GENERAL CLASS OF POLYNOMIALS Richard Andre-Jeannin IUT GEA, Route de Romain, 54400 Longwy, France (Submitted April 1994) 1. INTRODUCTION Let us consider the generalized Fibonacci polynomials U n (p,q; x) the generalized Lucas polynomials V n (p,q\ x) (or simply U V n if there is no danger of confusion) defined by U^Qc + pw^-qu^ (C7 0 = 0, ^ = 1), (1.1) V = (x + p)v^-qv n _ 2 (K 0 = 2,K 1 = x + p). (1.2) The parameters/? q as well as the variable x are arbitrary real numbers we denote by a = a(x) /? = J3(x) the numbers such that a + fl = x + p afi-q. The polynomials U n V n can be expressed by means of the Binet forms where Recall that a n -B n U n=^jir-> fora*0, (1.3) V n = a n +fi\ (1.4) A = A(x) = (x + pf-4q. (1.5) a = ((x + p) + A 1/2 )/2, f] = ((x + p)-a l/2 )/2. (1.6) Notice that A > 0 for every xif q <0 for all x sufficiently large if q > 0. Particular cases of U n (p,q;x) V n (p,q;x) are the Fibonacci Lucas polynomials (F (x) L n (x)), the Pell Pell-Lucas polynomials [6] (P n (x) <2 (x)), the first the second Fermat polynomials [7] (O n (x) @ (x)), the Morgan-Voyce polynomials [1, 2, 5, 8, 9, 10] (5 (x) C (x)), the Chebyschev polynomials (S (x) 7^(x)) given by U n (0 9-1; x) = F (x), F (0, - 1 ; x) = I (x) C/ (0-1; 2x) = P n (x), F w (0,-1; 2x) - &(x) / (0,2; x) = <D (x), F w (0,2; x) = 0 (x) (1.7) U n+l (2,l;x) = B n (xl C/ (0,1; 2x) - S w (x), K n (2,l;x) = C ll (x) F w (0,1; 2x) = 27 (x). In earlier papers [1, 2] the author has discussed the combinatorial properties of the coefficients of U n V n. Here, we shall investigate the differential properties satisfied by these polynomials, such as differential equations Rodrigues' formulas. 1995] 453
Let us define the sequence {c tk } n > k > 0 by c «,o = 2- rr, :»>0, (1.8) (2») Notice that ^ = ^ - ^ ± 1, n*k*l. (1.9) c, = 2 {2n)\n + k (n-k)\ Cn,k + i=(p 2 -k 2 Kk, n>k + l>\. (1.10) Our main results are the following theorems. Theorem 1: For every real number x, the polynomial the polynomial 7 7 < * - l ) -. * 77 > 1 " ~ k ~ l "' ' satisfy the differential equation E n y. y(k) = ay k>0 n k "' ' Theorem 2: For every x such that A > 0, we have Az" + (2k + l)(x + p)z' + (k 2 -n 2 )z = 0. (1.11) U =nc 0 A- m 4^A"- m, n>\, (1.12) where c w 0 is defined by (1.8). A!/2 " AW-1/2 r ^ ^ o A - ^ A ", «> 0, (1.13) More generally, we also have Rodrigues' formulas for U^k) V^k\ namely, Theorem 3: For every x such that A > 0 every k > 0, we have / 2 ; 2 \ rn-k-1 «" * "- k - 1 (1.14) F «= c w,, A - ^ / 2 ^ A " - 1 / 2, n>k, (1.15) where c w ^ is defined by (1.9). Notice that Theorem 3 reduces to Theorem 2 for k = 0 that (1.14) can be written, by (1.10), ^ = ^ A - * - 1 / 2 ^ ^ A " - 1 / 2, n>k. (1.16) 454 [NOV.
2. PROOF OF THEOREM 1 It is readily proven [3, 4] by (1.5) (1.6) that, for every x such that A > 0, \a' = aa- in, V ^'\ (21) thus that \(a n )' = na"a- 1 ' 2, \ (2.2) [( ")'= -n{]"a- in. By this, (1.3), (1.4), we see [3, 4] that K = nu (2.3) therefore that V^k)^nU {k ~ l \ k>\. (2.4) Notice that these identities are valid for every value of x, not only when A > 0, since the two members are polynomials. By (2.2), we also deduce that a n /5 n, whence V n = a" +/3" satisfies the differential equation (A m y) = n 2 A- m y, fora>0, (2.5) which is equivalent, for A > 0, to the equation 0 [see (1.11)], namely, Ay" + (x + p)y' -n 2 y = 0. (2.6) Notice that V n satisfies E n0 for every value of x, since, in that case, the first member of (2.6) is a polynomial. Differentiating (2.6) k times using Leibniz' rule, we see that z - y^ satisfies the differential equation E n%k (1.11). Hence, E n%k is satisfied by V< k \k>0, U {k ~ l) = ^ k \ k>\. This concludes the proof. For instance, the Morgan-Voyce polynomial B n {x)-u n+l {2X x ) equation n+1 x x(x + 4)z" + 3(x + 2)z f -n(n + 2)z = 0. satisfies the differential This result was first noticed by Swamy [10]. Remark: When A > 0, it is easy to verify that E tk can be written as la k+m z>] = (n 2 - k 2 )A k ~ V2 z, (2.7) which is a generalization of (2.5). We now give another (nonpolynomial) solution of E n^k. Proposition 1: Let n k be two integers such that n + k-l>0. ^ r A"" 1/2 is a solution of E tk. Then, for A > 0, the function 1995] 455
Proof: It is easy to verify that, for A > 0, A" 1/2 is a solution of the differential equation Ay" - {2n - 3)(x + p)y - {2n -\)y = 0. (2.8) Differentiating (2.8) {n + k-1) times putting z = y^n+k ~~ 1 \ we obtain Az" + 2(" + i~ l \x + p)z' + 2( n + *~ l \z-{2n-3)\ \{x + p)z'+\^ l J -{2n-l)z = 0. After some rearrangement, one can see that this equation is identical to E n k. Remark: Using the formulation (2.7) of E n k putting z d_ jn+k tffr,n+jc Changing A to (- -1) in (2.9), where n-k>2,we d_ K-k-M2 jn-k-\ = (rf-k 2 )A* A/f-1/2 = {n 2 -{k + l) 2 )A #,n-k-l ' =, n+k _, A, one can write " K l jn+k-l J2 i r 2\kk-\l2 «A/l-l/2 (2.9) ^& j.w+fc-1 ' obtain a formula that we shall need later: 2\ A -k-3/2 In particular, changing n to («+1), putting = - 1, we get jn-k-2 1 n-k-2 AH-1/2 (2.10) _d_ il/2 7«+l A w+1 AW+1/2 rf" = {n + \YA- i,z^~a n+i, \ n>0. (2.11) 3. PROOF OF THEOREM 2 In the proof of Theorem 2, we shall need the following well-known readily proven result: V^=\[(x+p)V n + MJ n \ (3.1) By (1.8), formula (1.12) (resp. (1.13)) is clear if n = 1 (resp. n = 0 or» = 1). Supposing that (1.12) (1.13) are true for n> 1, we get by (3.1) that K +i = n\ _ Al/2 (2/i!) On the other h, one can notice by (1.5) that (* + ^ A «- - + ^ > - - ase" " (3.2) " U Kn-\I1 = (2#i + l) {x+p)^-a n - i,l +n-^^a V F) " n ~ l n-l/2 a (3.3) From (3.2) (3.3), we see that "n+l ~ which is the needed formula for K w+l- n\ Ai/ 2 1 d w+l AW+1/2 (/i + l)! (2w!) 2n+\',«+l = 2 (2«-f2)!" \l/2 //7+1 ^,«+l A»+l/2 (3.4) 456 [NOV.
Now we see, by (2.3) (3.4), that U n+l -,,, V n+l ~ Z #i + l ^ (2n + 2)\ \l/2 a AW+1/2 *#.w+1 = 2 ( ^ ( W + 1)2A_1/2 S A " +1/2 ' by(2 - U) ' This completes the proof of Theorem 2. V ; (2«+ 2)! A" 4. PROOF OF THEOREM 3 We proceed by induction on k. By Theorem 2, statement (1.14) clearly holds for k - 0 every n>\. Supposing that (1.14) holds for k >0 every n>k + l we get, by (1.16),, by (2.10), we have at once that jn-k-1 Uf+1) = _!&)=: /y-k-l/2 _^ n \ ' n-k-l A«-1/2 A which is the needed formula for /^+1). On the other h, statement (1.15) holds for k - 0, by Theorem 2. When & > 1 we get, by (2.4) (1.14) that This completes the proof of Theorem 3. V& = nut l) = c^a~* +1/2 ^ A"" 1/2, n>k. REFERENCES 1. R. Andre-Jeannin. "A Note on a General Class of Polynomials." The Fibonacci Quarterly 32.5(1994):445-54. 2. R. Andre-Jeannin. "A Note on a General Class of Polynomials, Part II." The Fibonacci Quarterly 33.4 (1995):341-51. 3. P. Filipponi & A. F. Horadam. "Derivative Sequences of Fibonacci Lucas Polynomials." In Applications of Fibonacci Numbers 4:99-108. Ed. G. E. Bergum, A. N. Philippou, & A. F. Horadam. Dordrecht: Kluwer, 1991. 4. P. Filipponi & A. F. Horadam. "Second Derivative Sequences of Fibonacci Lucas Polynomials." The Fibonacci Quarterly 313 (1993): 194-204. 5. G. Ferri, M. Faccio, & A. D'Amico. "The DFF DFFz Triangles Their Mathematical Properties." In Applications of Fibonacci Numbers 5. Ed. A. N. Philippou, G. E. Bergum, & A. F. Horadam. Dordrecht: Kluwer, 1994. 6. A. F. Horadam & Br. J. M. Mahon. "Pell Pell-Lucas Polynomials." The Fibonacci Quarterly 23.1 (1985):7-20. 1995] 457
7. A. F. Horadam. "Chebyschev Fermat Polynomials for Diagonal Functions," The Fibonacci Quarterly 19.4 (1979):328-33. 8. J. Lahr. "Fibonacci Lucas Numbers the Morgan-Voyce Polynomials in Ladder Networks in Electrical Line Theory." In Applications of Fibonacci Numbers (AU? Please check Volume No.), pp. 141-61. Ed. A. N. Philippou, G. E. Bergum, & A. F. Horadam. Dordrecht: Kluwer, 1986. 9. M. N. S. Swamy. "Properties of the Polynomials Defined by Morgan-Voyce," The Fibonacci Quarterly 4.1 (1966): 73-81. 10. M. N. S. Swamy. "Further Properties of Morgan-Voyce Polynomials." The Fibonacci Quarterly 6.2 (1968): 167-75. AMS Classification Numbers: 11B39, 26A24, 11B83 > Author Title Index The 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 29150 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. D O N O T SEND PAYMENT W I T H 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 axe completed. 458 [NOV.