An algebraic approach to Lidstone polynomials

Abstract

A new definition of Lidstone polynomials [G.L. Lidstone, Note on the extension of Aitken's theorem (for polynomial interpolation) to the Everett types, Proc. Edinb. Math. Soc. 2 (2) (1929) 16-19] is proposed; this is a Hessemberg determinantal form. The algebraic approach provides an elementary proof of the main recursive properties of Lidstone polynomials.

Odd and even Lidstone-type polynomial sequences. Part 1: basic topics

“…Relation (21) can be considered as an infinite linear system in the unknowns p k (x). By solving the first k + 1 equations of this system by Cramer's rule, we get (23). Vice versa, if (23) holds, by expanding the determinant with respect to the last column [16], we have (22) and then the result follows from Theorem 2.…”

“…Here we do not give all the properties of classic Lidstone polynomials which can be obtained from the results of the previous sections and from a wide literature (see [5,11,15,23,27,36,37] and the references therein).…”

“…Some applications to quadrature formulas have been given too. Moreover, Costabile et al [23] introduced an algebraic approach to Lidstone polynomials {Λ k } k and a first form of the generalization of this sequence was proposed in [20,21].…”

Finding Determinant Forms of Certain Hybrid Sheffer Sequences

“…A vast literature associated with the matrix and other approaches to several special polynomials and corresponding hybrid forms can be found, see [8][9][10][11][12][13][14][15][16][17]. These matrix forms helps in solving various algorithms and in finding the solution of numerical and a general linear interpolation problems.…”

Eigenvalues of complementary Lidstone boundary value problems

“…Further characterization can be found in the study of [9][10][11][12][13][14][15][16]. More research on Lidstone interpolation as well as Lidstone spline is seen in [1,[17][18][19][20][21][22][23]. On the other hand, the Lidstone boundary value problems and several of its particular cases have been the subject matter of numerous investigations, see [4,18,[24][25][26][27][28][29][30][31][32][33][34][35][36][37] and the references cited therein.…”