Some results for sums of products of Chebyshev and Legendre polynomials

Abstract

In this paper, we perform a further investigation of the Gegenbauer polynomials, the Chebyshev polynomials of the first and second kinds and the Legendre polynomials. By making use of some analytic and combinatorial methods, we establish some new expressions for sums of products of arbitrary numbers of Chebyshev polynomials of the first and second kinds and Legendre polynomials.

Some Identities on Sums of Finite Products of the Pell, Fibonacci, and Chebyshev Polynomials

“…In this section, the main results of the paper on the sums of finite products of the Pell polynomials, Chebyshev polynomials of third and fourth kind, Fibonacci and Lucas numbers, and the derivative of the Pell polynomials are obtained using elementary computations. These results are established along the lines of the sums of finite products in Eqns ( 5)- (7) and are encapsulated in the following theorems.…”

“…Many authors have studied the properties of Chebyshev polynomials. For example, Zhang investigated the sums of finite products of the second kind of Chebyshev polynomials (7) and derived many identities, particularly…”

Some Identities and Inequalities Involving Symmetry Sums of Legendre Polynomials

“…These polynomials play a vital role in the study of function orthogonality and approximation theory, as a result, some scholars have dedicated themselves to studying their various natures and obtained a series of meaningful research results. The studies that are concerned with this content can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. Recently, Shen Shimeng and Chen Li [3] give certain symmetry sums of P n (x), and proved the following result:…”