Selçuk J. Appl. Math. Selçuk Journal of Special Issue. pp. 73-80, 2010 Applied Mathematics
The Algebraic Properties of Horadam Polynomials Necati Ta¸skara, Kemal Uslu, H. Hüseyin Güleç
Selçuk University, Science Faculty, Department of Mathematics, 42003, Kampus, Konya, Türkiye
e-mail: ntaskara@ selcuk.edu.tr,kuslu@ selcuk.edu.tr, hhgulec@ selcuk.edu.tr
Presented in 2National Workshop of Konya Ere˘gli Kemal Akman College, 13-14 May 2010.
Abstract. In this study, it is given new algebraic properties related to Horadam polynomials. Also, we obtain some alternative sums for these polynomials and investigate relations among other polynomials.
Key words: Horadam polynomials, Binomial sums. 2000 Mathematics Subject Classification: 11B83, 11C08. 1. Introduction
There are many studies on the properties of Fibonacci, Lucas sequences and their polynomials in literature [1], [3-5].Horadam polynomials have been also known as a generalization of many polynomials. In [2], these polynomials has been defined and given some their properties. In this study, we investigate new properties as binomial, sums and derivation for these polynomials.
Definition 1. [2]For ∈ Z the sequence of Horadam polynomials ( ; ) or () are defined the recurrence relation
(1) () = −1() + −2() ≥ 3
with initial conditions 1() = 2() = The characteristic equation of (1) is
2− − = 0 and its roots are
where 1 = √−
22+4 2 =
− √
22+4. Also, the binomial sum of Horadam
polynomial is (3) +1() = b 2c P =0 µ − ¶ ()−2+ + µ − ¶ b−1 2 c P =0 µ − − 1 ¶ ()−2
2. The Some Algebraic Properties of Horadam polynomials
In this section, we obtain the Horadam polynomials as binomial sums and give new some algebraic properties for these polynomials.
Theorem 1. Let () be Horadam polynomial. Then, we have
(4) () = 2−2 b−2 2 c P =0 µ − 1 2 + 1 ¶ ()−2−2¡22+ 4¢ +2−3 b−3 2 c P =0 µ − 2 2 + 1 ¶ ()−3−2¡22+ 4¢
Proof. By using the Binet formula, we have () = −1+ −1 = ( − ) −1− ( − ) −1 − = µ −1− −1 − ¶ + µ −2− −2 − ¶ = − Ã +p22+ 4 2 !−1 − − Ã −p22+ 4 2 !−1 + − Ã +p22+ 4 2 !−2 − − Ã −p22+ 4 2 !−2
If we arrange the above last equality using Binomial expansion, then we obtain () = − −1 X =0 µ − 1 ¶ ³ 2 ´−1−Ã p22+ 4 2 ! − − X−1 =0 µ − 1 ¶ ³ 2 ´−1−Ã p22+ 4 2 ! (−1) + − X−2 =0 µ − 2 ¶ ³ 2 ´−2−Ã p22+ 4 2 ! − − X−2 =0 µ − 2 ¶ ³ 2 ´−2−Ã p22+ 4 2 ! (−1) From where, we can write
() = 2−2 b−2 2 c X =0 µ − 1 2 + 1 ¶ ()−2−2¡22+ 4¢ + 2−3 b−3 2 c X =0 µ − 2 2 + 1 ¶ ()−3−2¡22+ 4¢
The following theorems give us the sums corresponding different indexes of Horadam polynomials.
Theorem 2. Let () denote the th Horadam polynomial. Then, the sum of even indexes of these polynomials is
X =1 2() = 22() − 2+2() + + − 2− 22− 2 + 1
Proof. From equation (2), we can write X =1 2() = X =1 " Ã 2−1− 2−1 − ! + Ã 2−2− 2−2 − !# = − " 1 X =1 2− 1 X =1 2 # + − " 1 2 X =1 2− 1 2 X =1 2 #
If we make sufficient arrangements, then we have P =1 2() = ¡ 2− 2− 2+ 1¢ ∙ 2 µ 2−1− 2−1 − ¶ − µ 2+1− 2+1 − ¶ − + 1 ¸ +¡ 2− 2− 2+ 1¢ ∙ 2 µ 2−2− 2−2 − ¶ − µ 2− 2 − ¶ + 2− 2 − ¸ By considering the Binet formula, we conclude
X =1 2() = 1 (2− 22− 2 + 1) £ 22() − 2+2() − + + ¤
Theorem 3. Let () denote th Horadam polynomial. Then, the sum of odd indexes of these polynomials is
X =1 2+1() = 22+1() − 2+3() + 2− 2+ (2− 22− 2 + 1)
Proof. If we use the Binet formula, then it can be written X =1 2+1() = X =1 " Ã 2− 2 − ! + Ã 2−1− 2−1 − !# = − " X =1 2− X =1 2 # + − " 1 X =1 2− 1 X =1 2 # By considering the property of geometric sum, it is obtained
P =1 2+1() = − h³ 2+2 −1 2−1 − 1 ´ −³2+22 −1 −1 − 1 ´i +−h1 ·³2+22−1−1 − 1 ´ −1· ³ 2+2 −1 2−1 − 1 ´i
If we make sufficient arrangements, then we have X =1 2+1() = ¡ 2− 2− 2+ 1¢ ∙ 2 µ 2− 2 − ¶ − µ 2+2− 2+2 − ¶ + µ 2− 2 − ¶¸ +¡ 2− 2− 2+ 1¢ ∙ 2 µ 2−1−2−1 − ¶ − µ 2+1−2+1 − ¶ − + 1 ¸ = 1 (2−22−2+1) £ 2 2+1() − 2+3() + 2− 2+ ¤
The following results give us the sums corresponding different indexes of Ho-radam polynomials as a generalization of above theorems.
Theorem 4. For Horadam polynomial () we have X =1 +() = (−) +() − (−)() − ++() + +() (−)− − + 1
Proof. From the Binet formula, we can write X =1 +() = X =1 " Ã +−1− +−1 − ! + Ã +−2− +−2 − !# = − " X =1 −1− X =1 −1 # + − " X =1 −2− X =1 −2 # By considering the geometric sum, it is obtained
P =1 +() = − h −1³+ −1 −1 − 1 ´ − −1³+−1−1− 1 ´i +−h−2³+ −1 −1 − 1 ´ − −2³+−1−1 − 1 ´i If it is made sufficient arrangements, then we have
where = (− 1)( − 1) P =1 +() From (− 1)(− 1) = (−)− − + 1 and (2), we obtain X =1 +() = (−) +() − (−)() − ++() + +() (−)− − + 1 as required.
Theorem 5. The ratio of limits of consecutive terms is equal to the positive root of characteristic equation of (1).
Proof. From Definition 1 and (2), we have lim →∞ +1() () = lim →∞ ( − ) − ( − ) ( − ) −1− ( − ) −1 = lim →∞ ( − ) − ( − )³ ´ ( − )1 − ( − ) 1 ³ ´ =
Theorem 6. The Horadam polynomial () can be written as linear combi-nation of first order derivations such that
−1() = 1 ¡
0() − 0−1() − 0−2()¢
Proof. If it is derivatived in equation (1), then one can see the result −1() =
1 ¡
0() − 0−1() − 0−2()¢
Theorem 7. The derivation of Horadam polynomials () with binomial coefficients is 0+1() = b−1 2 c X =0 µ − ¶ ( − 2) −1−2−2 + µ − ¶ b−2 2 c X =0 µ − − 1 ¶ ( − 2) −1−2−2
Proof. From equation (3), we can write +1() = b 2c P =0 µ − ¶ ()−2+ µ − ¶ b−1 2 c P =0 µ − − 1 ¶ ()−2 Taking derivation of both sides of last equation, we have
0+1() = b−1 2 c X =0 µ − ¶ ( − 2) −1−2−2 + µ − ¶ b−2 2 c X =0 µ − − 1 ¶ ( − 2) −1−2−2
Similarly, it is also obtained 2th order derivation of Horadam polynomials such that 00 +1() = b−2 2 c P =0 µ − ¶ ( − 2) ( − 1 − 2) −2−2−2 +³ − ´ b−3 2 c P =0 µ − − 1 ¶ ( − 2) ( − 1 − 2) −2−2−2
By considering the recurrence relation (1) of Horadam polynomials, it is given first and second order derivations in the following table:
1() = 0 1() = 0 00 1() = 0 2() = 0 2() = 00 2() = 0 3() = 2+ 0 3() = 2 00 3() = 2 4() = 23+ + 0 4() = 322+ + 00 4() = 62 .. . ... ... Table 1:
From Table 1, the coefficients of first order derivations are given as follows: 2 32 → 43 → 54 → 33+ 92 65 → 44+ 163 .. . ... ... + 22+ 4 → 22 + 2 → 622+ 62 .. . ... Table 2:
Similarly, the coefficients of second order derivations are also given as follows:
2 62 123 → 204 → 305 → 124+ 483 .. . ... ... 22+ 4 63+ 182 → 622+ 62 .. . ... Table 3:
Consequently, all theorems given in this study give us well-known Fibonacci, Lucas, Pell, Pell-Lucas, Jacobsthal, Jacobsthal-Lucas sequences and their poly-nomials for proper values of and in literature. Thus, obtained results are generalizations of some sequences and polynomials.
References
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