Research Article | Open Access
Sequences of Non-Gegenbauer-Humbert Polynomials Meet the Generalized Gegenbauer-Humbert Polynomials
Here, we present a connection between a sequence of polynomials generated by a linear recurrence relation of order 2 and sequences of the generalized Gegenbauer-Humbert polynomials. Many new and known transfer formulas between non-Gegenbauer-Humbert polynomials and generalized Gegenbauer-Humbert polynomials are given. The applications of the relationship to the construction of identities of polynomial sequences defined by linear recurrence relations are also discussed.
Many number and polynomial sequences can be defined, characterized, evaluated, and classified by linear recurrence relations with certain orders. A polynomial sequence is called a sequence of order 2 if it satisfies the linear recurrence relation of order 2 for some coefficient and and initial conditions and . To construct an explicit formula of its general term, one may use a generating function, characteristic equation, or a matrix method (see Comtet , Hsu , Strang , Wilf , etc.). In , the authors presented a new method to construct an explicit formula of generated by (1.1). For the sake of the reader's convenience, we cite this result as follows.
Proposition 1.1. Let be a sequence of order 2 satisfying the linear recurrence relation (1.1), then where and are roots of , namely,
In , Aharonov et al. have proved that the solution of any sequence of numbers that satisfies a recurrence relation of order 2 with constant coefficients and initial conditions and , called the primary solution, can be expressed in terms of Chebyshev polynomial values. For instance, the authors show and , where and are, respectively, Fibonacci numbers and Lucas numbers, and and are the Chebyshev polynomials of the first kind and the second kind, respectively. Some identities drawn from those relations were given by Beardon in . Marr and Vineyard in  use the relationship to establish explicit expression of five-diagonal Toeplitz determinants. In , the authors presented a new method to construct an explicit formula of generated by (1.1). Inspired with those results, in , The authors and Weng established a relationship between the number sequences defined by recurrence relation (1.1) and the generalized Gegenbauer-Humbert polynomial value sequences. The results are suitable for all such number sequences defined by (1.1) with arbitrary initial conditions and , which includes the results in [6, 7] as the special cases. Many new and known formulas of Fibonacci, Lucas, Pell, and Jacobsthal numbers in terms of the generalized Gegenbauer-Humbert polynomial values were presented in . In this paper, we will give an alternative form of (1.2) and find a relationship between all polynomial sequences defined by (1.1) and the generalized Gegenbauer-Humbert polynomial sequences.
A sequence of the generalized Gegenbauer-Humbert polynomials is defined by the expansion (see, e.g., , Gould , and the authors with Hsu ) where , and are real numbers. As special cases of (1.4), we consider as follows (see ): Chebyshev polynomial of the second kind, , Legendre polynomial, , Pell polynomial,, Fibonacci polynomial,, Morgan-Voyc polynomial,  by Koshy, , Fermat polynomial of the first kind,, Dickson polynomial of the second kind, (see, e.g., ) by Lidl et al.,
where is a real parameter, and is the Fibonacci number. In particular, if , the corresponding polynomials are called Gegenbauer polynomials (see ). More results on the Gegenbauer-type polynomials can be found in Hsu  and Hsu and Shiue , and so forth, it is interesting that for each generalized Gegenbauer-Humbert polynomial sequence there exists a nongeneralized Gegenbauer-Humbert polynomial sequence, for instance, corresponding to the Chebyshev polynomials of the second kind, Pell polynomials, Fibonacci polynomials, Fermat polynomials of the first kind, and the Dickson polynomials of the second kind, we have the Chebyshev polynomials of the first kind, Pell-Lucas polynomials (see  by Horadam and Mahon), Lucas polynomials, the Fermat polynomials of the second kind (see  by Horadam), and the Dickson polynomials of the first kind, respectively.
Similarly, for a class of the generalized Gegenbauer-Humbert polynomial sequences defined by for all with initial conditions the following theorem is obtained.
In next section, we will use an alternative form of (1.2) to establish a relationship between the polynomial sequences defined by recurrence relation (1.1) and the generalized Gegenbauer-Humbert polynomial sequences defined by (1.5). Many new and known formulas of polynomials in terms of the generalized Gegenbauer-Humbert polynomials and applications of the established relationship to the construction of identities of polynomial sequences will be presented in Section 3.
2. Main Results
We now modify the explicit formula of the polynomial sequences defined by linear recurrence relation (1.2) of order 2. If , the first formula in (1.2) can be written as Noting that , we may further write the above expression of as
Denote and . To find a transfer formula between expressions (1.7) and (2.2), we set for a nonzero real or complex-valued function , which are two roots of . Thus, adding and multiplying two equations of (2.3) side by side, we obtain The above system implies and at and give expressions of and as It is clear that and satisfy and .
Therefore, we obtain our main result.
Theorem 2.1. Let sequence be defined by with initial conditions and , then can be presented as where is the sequence of any generalized Gegenbauer-Humbert polynomials with . In particular, can be expressed in terms of , the sequence of the Chebyshev polynomials of the second kind, which is a special case of (2.10) for .
Corollary 2.2. For , respectively, from (2.10), one has transfer formulas where , , , , and are the Chebyshev polynomials of the second order, Pell polynomials, Fibonacci polynomials, Fermat polynomials, and the Dickson polynomials of the second kind, respectively.
Example 2.3. As the first example, we consider the Chebyshev polynomials of the first kind satisfying recurrence relation (1.1) with and and initial conditions and . From Corollary 2.2, we have
in which the first relation is equivalent to the well-known result due to
For the special cases of and , we have the following corollaries.
Corollary 2.4. Let sequence be defined by () with initial conditions and . Then
Corollary 2.5. Let sequence be defined by with initial conditions and , then
Corollary 2.6. Let sequence be defined by with initial conditions and .
Then In addition, one has where are the Chebyshev polynomials of the first kind.
Proof. It is sufficient to prove the positive case of (2.18). From the first formula shown in Corollary 2.6 and the recurrence relation , one easily sees From the first formula of Example 2.3, the above last expression of implies the positive case of (2.18). The negative case can be proved similarly.
Example 2.7. As an example, the Lucas polynomial sequence defined by (1.1) with and and initial conditions and has an explicit formula for its general term
Using Corollary 2.6, we also have
From Theorem 2.1, one may obtain transfer formulas between generalized Gegenbauer-Humbert polynomials.
3. Examples and Applications
Example 3.1. The Chebyshev polynomials of the third kind and fourth kind satisfy the same recurrence relationship as the Chebyshev polynomials of the first kind with the same constant initial term 1 and different linear initial terms, and , respectively (see, e.g., Mason and Handscomb  and Rivlin ). Hence, the Chebyshev polynomials of the third kind, , and the Chebyshev polynomials of the fourth kind, , when , have the following expressions using the argument shown in :
Similarly to the Chebyshev polynomials of the first kind (see Example 2.3), we can transfer and to the generalized Gegenbauer-Humbert polynomials with , From the above formulas, one may obtain some identities between the Chebyshev polynomials of different kinds. For instance,
Since , , , and , where , the above identities of Chebyshev polynomials also present the following identities of trigonometric functions, respectively,
Example 3.2. Consider the Jacobsthal polynomials defined by (1.1) with coefficients and and initial conditions . One may use Corollary 2.5 to obtain transfer formulas The first formula and its inverse (see the first formula below) were given on [20, page 76] by Riordan using a different method. The positive case of the third formula is easily to be transferred to the formula of Theorem 1 in , where they used a different recurrence relation with and for constructing the Jacobsthal polynomials. Reference  also gave the inverse formula to present in terms of . Actually, we can easily have the inverse formulas of , , , , and in terms of as follows:
Example 3.3. In Eu , the polynomial sequence is defined by with initial conditions and . Using Corollary 2.5, we obtain in which the first formula was given in  using a different approach. Similar to the case of the Jacobsthal polynomial sequence shown in Example 3.2, we have the inverse formulas
Another polynomial sequence is defined by with initial conditions and . Using Corollary 2.5, we obtain where the first formula has been established in  by using a different method. The inverse of the above formulas can be found similarly. For instance,
Example 3.4. In Riordan , the associate Legendre polynomial sequence is defined by with initial conditions and , then we use Theorem 2.1 and Corollary 2.2 to generate the following transfer formulas: where the first formula was given on [20, page 85] using a different method.
Example 3.5. In Chow and West , the polynomial sequence is defined by with initial conditions and (). From Theorem 2.1 and Corollary 2.2, we obtain
Since , we have
Hence, from the last expression of and the transfer formula of in terms of shown above, we obtain
in which the case of
was established in  using mathematical induction.
Equaling the right-hand expressions of the polynomials shown in each example, one may obtain various identities of generalized Gegenbauer-Humbert polynomials. For instance, from Example 2.3, we have
Using the relationship established in Theorem 2.1 and Corollaries 2.2–2.6, we may obtain some identities of polynomial sequences from the generalized Gegenbauer-Humbert polynomial sequence identity described in  where satisfies the recurrence relation of order 2, with coefficients and , and and . Clearly (see (2.41) in ), For , we have , where are the Chebyshev polynomials of the second kind, and we can write (3.17) as where and . From the first formula of Example 3.2 and using transform , we have Substituting the above expression to (3.19) yields the identity Similarly, from Example 3.3, we obtain identities
One may also extend some well-known identities of a polynomial sequence to other polynomial sequences using the relationships we have established. For instance, from the Cassini-like formula for Fibonacci polynomials we use the relationship shown in Example 3.2 to obtain the Cassini-like formula for the Jacobsthal polynomials which can be transferred to the formula of Theorem 2 in  using the same argument in Example 3.2.
Similarly, from the transform we have
To construct a transform relationship for the polynomials defined by recurrence relation with coefficients related to the order of polynomials is much more difficulty. One special example can be found on [25, page 240] by Andrews et al.. It seems there is no a general method applied to such polynomial sequences.
The authors wish to thank the referees and editors for their helpful comments and suggestions.
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