FIBONACCI AND LUCAS DIFFERENTIAL EQUATIONS ESRA ERKU¸ S -DUMAN AND HAKAN CIFTCI Abstract. The second order linear hypergeometric di¤erential equation and the hypergeometric function play a central role in many areas of mathematics and physics. The purpose of this paper is to obtain di¤erential equations and the hypergeometric forms of the Fibonacci and the the Lucas polynomials. We also write again these polynomials by means of Olver’s hypergeometric functions. In addition, we present some relations between these polynomials and the other well-known functions.
1. Introduction In modern science there is a huge interest in the theory and application of the Fibonacci and the Lucas numbers [8, 10, 11]. The Fibonacci and the Lucas polynomials are also important in a wide variety of research subjects [3, 6, 9, 14, 16, 17]. Large classes of polynomials can be de…ned by Fibonacci-like recurrence relation. Fibonacci polynomials were studied in 1883 by the Belgian mathematician Eugene Charles Catalan and the German mathematician E. Jacobsthal. The polynomials Fn (x) studied by Catalan are de…ned by the recurrence relation (1.1)
Fn (x) = xFn
1
(x) + Fn
2
(x) ; n
3;
where F1 (x) = 1, F2 (x) = x. These polynomials also yield recurrence relation n+1 x 0 (1.2) Fn+1 (x) = Fn (x) + Fn0 (x) : 2 2 The Lucas polynomials Ln (x), originally studied in 1970 by Bicknell, are de…ned by (1.3)
Ln (x) = xLn
1
(x) + Ln
2
(x) ; n
2;
where L0 (x) = 2, L1 (x) = x. These polynomials also yield recurrence relation i n+1h x (1.4) L0n+1 (x) = Ln (x) + L0n (x) : 2 n In 1769, Euler formed the hypergeometric di¤erential equation of the form [5] (1.5)
x (1
x) y 00 + [c
(a + b + 1) x] y 0
aby = 0
which has three regular singular points: 0; 1 and 1. The hypergeometric di¤erential equation is a prototype: Every ordinary di¤erential equation of second-order with at most three regular singular points can be brought to the hypergeometric di¤erential equation by means of a suitable change of variables. The solutions of hypergeometric di¤erential equation include many of the most interesting special functions of mathematical physics [2, 13, 15]. Solutions of the Key words and phrases. Fibonacci polynomial, Lucas polynomial, recurrence relation, Gaussian function, hypergeometric di¤erential equation. 1
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ESRA ERKU S ¸ -DUM AN AND HAKAN CIFTCI
hypergeometric di¤erential equation are built out of the hypergeometric series.The solution of Euler’s hypergeometric di¤erential equation is called hypergeometric function or Gaussian function 2 F1 introduce by Gauss [7] as follows: (1.6)
2 F1
(a; b; c; x) = F (a; b; c; x) =
1 X (a)k (b)k
k=0
(c)k k!
xk ;
whose natural generalization of an arbitrary number of p numerator and q denominator parameters (p; q 2 N0 := N[ f0g) is called and denoted by the generalized hypergeometric series p Fq de…ned by p Fq
1 ; :::;
p;
1 ; :::;
q;
z =
1 X (
k=0
=
::: ( ( 1 )k ::: (
p Fq
1 )k
(
1 ; :::;
zk q )k k! p )k
p;
1 ; :::;
q; z) :
Here ( ) denotes the Pochhammer symbol de…ned (in terms of gamma function) by ( ) = =
( + ) ( 2 C n Z0 ) ( ) 1; ( + 1):::( + n 1);
if if
= 0; 2 Cnf0g : = n 2 N; 2 C
and Z0 denotes the set of nonpositive integers and ( ) is the familiar Gamma function. In this paper, we obtain a di¤erential equation for the Fibonacci polynomials and the hypergeometric form of these polynomials via the hypergeometric di¤erential equation and the Gaussian function. Similarly, we get a di¤erential equation for the Lucas polynomials and the hypergeometric form of these polynomials. Finally, we derrive the Fibonacci and the Lucas polynomials in terms of the Olver’s hypergeometric functions and give some relations between these polynomials and some known functions such as Legendre, Gegenbauer, Jacobi functions. 2. Fibonacci Polynomials In this section, we obtain Fibonacci di¤erential equation and the hypergeometric form of the Fibonacci polynomials. Then, we give some relations between the Fibonacci polynomials and associated Legendre functions, Gegenbauer functions, Jacobi functions respectively. Theorem 2.1. The Fibonacci polynomials Fn (x) satisfy the di¤ erential equation (2.1)
x2 + 4 y 00 + 3xy 0
n2
1 y = 0:
Proof. Combining the recurrence relations (1.1) and (1.2) for the Fibonacci polynomials Fn (x) and after some calculations, we arrive at the desired result. Theorem 2.2. The Fibonacci polynomials Fn (x) can be written by the hypergeometric function as follows: (2.2)
Fn (x) =
2 F1
1
n 1+n 3 x2 ; ; ;1 + 2 2 2 4
:
3
Proof. Because the Fibonacci polynomials Fn (x) is the solution of the equation (2.1), we can write 1+
x2 4
00
Fn +
Here, if we use the mapping z = 1 + 2
z (1
z)
d Fn + dz 2
3x 0 F 4 n
x2 4 ,
3 2
n2
1 4
Fn = 0:
then we get dFn n2 1 + Fn = 0: dz 4
2z
Comparing the last equation with the hypergeometric di¤erential equation (1.5), we obtain the constants 1+n 3 1 n ; b= ; c= : a= 2 2 2 Thus, in view of solutions of the hypergeometric di¤erential equation are the hypergeometric functions 2 F1 , we get Fn (z) =2 F1 Now, writing again z = 1 +
x2 4
1 2
n 1+n 3 ; ; ;z : 2 2
in the solution, the proof is completed.
2 F1 (a; b; c; z) is the standart notation for the principal solution of the hypergeometric di¤erential equation (1.5). But it is more convenient to develop the theory in terms of the function 2 F1 (a; b; c; z) ; (2.3) F (a; b; c; z) = (c)
because this leads to fewer restrictions and simpler formulas. From (1.6) and (2.3), we have 1 X (a)k (b)k z k F (a; b; c; z) = ; (jzj < 1) ; (c + k) k! k=0
which is known as Olver’s hypergeometric function which is de…ned by Olver [12]. Olver also obtained various relations between these polynomials and other known functions. Two of them are 1
(2.4) (2.5)
F (a; 1 F a; b;
a; c; z) =
a+b+1 ;z 2
z 1
= ( z (1
c 2
P 1 a c (1
z z))
1
a 4
b
1
Pa
2z) ; a b 2 b 1 2
(1
2z) ;
where P (z) is the associated Legendre function [1]. This function is a solution of the associated Legendre equation 1
z2
d2 y dz 2
2z
dy + dz
2
( + 1)
1
z2
y = 0;
which is the importance in various branches of applied mathematics and physics, particularly the solution of Laplace’s equation in spherical polar or spheroidal coordinates. On the other hand, any hypergeometric function for which a quadratic transformation exist can be expressed in terms of associated Legendre functions. So, we can write the following relations between these functions and other known functions:
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ESRA ERKU S ¸ -DUM AN AND HAKAN CIFTCI
(2.6)
2
P (x) =
(1
(
2 ) ( +
+ 1)
) (x2
+ 1) (1
1)
=2
C
( 21 +
)
(x) ;
where the Gegenbauer function C (x) is a generalization of the Gegenbauer (or ultraspherical) polynomials and has the representation [4] ( +2 ) 2 F1 ( + 1) (2 )
C (x) =
(2.7)
P (x) =
1 (1
x+1 x 1
)
( ; )
where the Jacobi function and has the representation [1] ( ; )
(x) =
2 F1
1 ( + 2
1 1 x ; + ; 2 2
+2 ;
=2 (
; ) i(2 +1)
arcsin h
x
: 1
2
1=2
!!
;
(x) is a generalization of the Jacobi polynomials
+1
i );
1 ( + 2
+ 1 + i );
+ 1;
sinh2 t :
We can write the Fibonacci polynomials Fn (x) in terms of Olver’s hypergeometric function as follows: p 1 n 1+n 3 x2 (2.8) Fn (x) = F ; ; ;1 + : 2 2 2 2 4 Now, we have the following theorem which gives relations between Fibonacci polynomials Fn (x) and the associated Legendre functions, the Gegenbauer functions, the Jacobi functions respectively: Theorem 2.3. For the Fibonacci polynomials Fn (x), we have p 1 x x2 2 P (i) Fn (x) = 1 n 1 1=4 2 2 2 (x2 + 4) x x2 (ii) Fn (x) = C 1n2 1 1 n 2 !! r x2 ( 12 ; 12 ) (iii) Fn (x) = in arcsin h i 1 + : 4 Proof. (i) Using Olver’s hypergeometric form of the Fibonacci polynomials (2.8) in relation (2.4), we arrive at this result. (ii) Taking into consideration relations (2.4), (2.6) and (2.8), we obtain the desired result. (iii) Taking into consideration relations (2.4), (2.7) and (2.8), we obtain the desired result. 3. Lucas Polynomials In this section, we obtain Lucas di¤erential equation and the hypergeometric form of the Lucas polynomials. Then, we give some relations between the Lucas polynomials and associated Legendre functions, Gegenbauer functions, Jacobi functions respectively.
5
Theorem 3.1. The Lucas polynomials Ln (x) satisfy the di¤ erential equation x2 + 4 y 00 + xy 0
n2 y = 0:
Proof. Combining the recurrence relations (1.3) and (1.4) for the Lucas polynomials Ln (x) and after some calculations, we arrive at the desired result. Theorem 3.2. The hypergeometric form of the Lucas polynomials can be written as follows: n n 1 x2 Ln (x) = 2 F1 ; ; ;1 + : 2 2 2 4 Proof. For the Lucas polynomials Ln (x), we can write 1+
x2 4
00
Ln +
Here, if we use the mapping z = 1 + z (1
z)
d2 Ln + dz 2
x2 4 ,
1 2
x 0 L 4 n
n2 Ln = 0: 4
then we get z
dLn n2 + Ln = 0: dz 4
Comparing the last equation with the hypergeometric di¤erential equation (1.5), we obtain the constants n 1 n a= ; b= ; c= : 2 2 2 Thus, in view of solutions of the hypergeometric di¤erential equation are the hypergeometric functions 2 F1 , we get Ln (z) =2 F1 Now, writing again z = 1 +
x2 4
n n 1 ; ; ;z : 2 2 2
in the solution, the proof is completed.
We can write the Lucas polynomials Ln (x) in terms of Olver’s hypergeometric function as follows: p x2 n n 1 (3.1) Ln (x) = ; ; ;1 + : F 2 2 2 4 Now, we have the following theorem which gives relations between Lucas polynomials Ln (x) and the associated Legendre functions, the Jacobi functions respectively: Theorem 3.3. For the Lucas polynomials Ln (x), we have p 1 1 x 2 x2 (i) Ln (x) = x + 4 4 P 2 n+1 1 2 2 2 !! r x ( 21 ; 21 ) x2 (ii) Ln (x) = arcsin h i 1 + : 2 in 4 Proof. (i) Using Olver’s hypergeometric form of the Fibonacci polynomials (3.1) in relation (2.5), we arrive at this result. (ii) Taking into consideration relations (2.5), (2.7) and (3.1), we obtain the desired result.
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ESRA ERKU S ¸ -DUM AN AND HAKAN CIFTCI
References [1] M. Abramowitz and I. A. Stegun, (Eds.). "Legendre Functions." Ch. 8 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 331-339, 1972. [2] W. N. Bailey, Generalized Hypergeometric Series, Cambridge, England University Press, 1935. [3] G. B. Djordjevi´c, Some properties of partial derivatives of generalized Fibonacci and Lucas polynomials, Fibonacci Quart. 39.2 (2001) 138-141. [4] A. Erdélyi, W. Magnus, F. Oberhettinger and F. G. Tricomi, Higher Transcendental Functions, Vol. II, McGraw-Hill Book Company, New York, Toronto and London, 1955. [5] L. Euler, Institutiones Calculi Integralis, vol.1 of Opera Omnia Series, 1769. [6] E. Erkus-Duman and N. Tuglu, Generating functions for the generalized bivariate Fibonacci and Lucas polynomials, J. Comput. Anal. Appl. 18 (2015) 815-821. [7] C. F. Gauss, Disquisitiones Generales Circa Seriem In…nitam, vol.3, Werke, 1813. [8] K. Hawkins, U. Hebert-Johnson and B. Mathes, The Fibonacci identities of orthogonality, Linear Algebra Appl. 475 (2015) 80-89. [9] H. Kitayama and D. Shiomi, On the irreducibility of Fibonacci and Lucas polynomials over …nite …elds, Finite Fields and Their Applications, 48 (2017) 420-429. [10] T. Koshy, Fibonacci and Lucas Numbers with Applications, Toronto, New York, NY, USA, 2001. [11] G. -Y. Lee, k-Lucas numbers and associated bipartite graphs, Linear Algebra Appl. 320 (2000) 51-61. [12] F. W. J. Olver, Asymptotics and Special Functions, AKP Classics, 1997. [13] E. D. Rainville, Special Functions, The Macmillan Company, New York, 1960. [14] R. F. Remis, On the relation between FDTD and Fibonacci polynomials, J. Comput.Phys. 230 (2011) 1382-1386. [15] H. M. Srivastava and H. L. Manocha, A Treatise on Generating Functions, Halsted Press (Ellis Horwood Limited, Chichester), John Wiley and Sons, New York, 1984. [16] W. Wang and H. Wang, Generalized Humbert polynomials via generalized Fibonacci polynomials, Appl. Math. Comput., 307 (2017) 204-216. [17] X. Ye and Z. Zhang, A common generalization of convolved generalized Fibonacci and Lucas polynomials and its applications, 306 (2017) 31-37.
Esra Erku¸ s-Duman Gazi University, Faculty of Science, Department of Mathematics, Teknikokullar TR-06500, Ankara, Turkey. E-mail addresses:
[email protected],
[email protected] Hakan Ciftci Gazi University, Faculty of Science, Department of Physics, Teknikokullar TR-06500, Ankara, Turkey. E-mail address:
[email protected]
