Here, we present a connection between a sequence of polynomials generated by a linear recurrence rel

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1 Department of Mathematics and Computer Science, Illinois Wesleyan University, Bloomington, IL 61702, USA 2 Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV 89154-4020, additionally USA
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. 1. Introduction
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 ( ) = ( ) 1 + ( ) 2 ( ) , 2 , ( 1 . 1 ) for some coefficient ( ) 0 and ( ) 0 and initial additionally conditions 0 ( ) and 1 ( ) . To construct an explicit formula of its general term, one may use a generating function, characteristic equation, or a matrix method (see Comtet [ 1 ], Hsu [ 2 ], Strang [ 3 ], Wilf [ 4 ], etc.). In [ 5 ], 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 ( ) = 1 ( ) ( ) 0 ( ) ( ) ( ) ( ) 1 ( ) ( ) 0 ( ) ( ) ( ) ( ) i f ( ) ( ) , 1 ( ) 1 ( ) ( 1 ) 0 ( ) ( ) i f ( ) = ( ) , ( 1 . 2 ) where ( ) and ( ) are roots of 2 ( ) ( ) = 0 , namely, 1 ( ) = 2 ( ) + 2 1 ( ) + 4 ( ) , ( ) = 2 ( ) 2 . ( ) + 4 ( ) ( 1 . 3 )
In [ 6 ], 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 0 = 0 and 1 = 1 , called the primary solution, can be expressed in terms of Chebyshev polynomial values. For instance, the authors show = ( / 2 ) and = 2 ( / 2 ) , where and are, respectively, Fibonacci numbers and Lucas numbers, and ( ) and ( ) are the Chebyshev polynomials additionally of the first kind and the second kind, respectively. Some identities drawn from those relations were given by Beardon additionally in [ 7 ]. Marr and Vineyard in [ 8 ] use the relationship to establish explicit expression of five-diagonal Toeplitz determinants. In [ 5 ], the authors presented a new method to construct an explicit formula of { ( ) } generated by ( 1.1 ). Inspired with those results, in [ 9 ], 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 additionally sequences defined by ( 1.1 ) with arbitrary initial conditions 0 and 1 , which includes the results in [ 6 , 7 ] as the special additionally 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 [ 9 ]. In this paper, we will give an alternative form of ( 1.2 ) and find a relationship additionally between additionally all polynomial sequences defined by ( 1.1 ) and the generalized Gegenbauer-Humbert polynomial sequences.
A sequence of the generalized Gegenbauer-Humbert polynomials { , , ( ) } 0 is defined by the expansion (see, e.g., [ 1 ], Gould [ 10 ], and the authors with Hsu [ 11 ]) Φ ( ) 2 + 2 = 0 , , ( ) , ( 1 . 4 ) where > 0 , and 0 are real numbers. As special cases of ( 1.4 ), we consider , , ( ) as follows (see [ 11 ]): 1 , 1 , 1 ( ) = ( ) , Chebyshev polynomial of the second kind, 1 / 2 , 1 , 1 ( ) = ( ) , Legendre polynomial, 1 , 1 , 1 ( ) = + 1 ( ) , Pell polynomial, 1 , 1 , 1 ( / 2 ) =