Hankel determinants of sequences related to Bernoulli and Euler polynomials

We evaluate the Hankel determinants of various sequences related to Bernoulli and Euler numbers and special values of the corresponding polynomials. Some of these results arise as special cases of Hankel determinants of certain sums and differences of Bernoulli and Euler polynomials, while others are consequences of a method that uses the derivatives of Bernoulli and Euler polynomials. We also obtain Hankel determinants for sequences of sums and differences of powers and for generalized Bernoulli polynomials belonging to certain Dirichlet characters with small conductors. Finally, we collect and organize Hankel determinant identities for numerous sequences, both new and known, containing Bernoulli and Euler numbers and polynomials.

[1]  Christian Krattenthaler,et al.  Some determinants of path generating functions, II , 2009, Adv. Appl. Math..

[2]  Guo-Niu Han.  Hankel continued fractions and Hankel determinants of the Euler numbers , 2019, Transactions of the American Mathematical Society.

[3]  Stephen C. Milne,et al.  Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions , 2000, math/0008068.

[4]  Lin Jiu,et al.  Orthogonal polynomials and Hankel determinants for certain Bernoulli and Euler polynomials , 2020, Journal of Mathematical Analysis and Applications.

[5]  Kwang-Wu Chen.  A summation on Bernoulli numbers , 2005 .

[6]  C. Jordan.  Advanced Determinant Calculus: a Complement , 2008 .

[7]  Nombres de q-Bernoulli-Carlitz et fractions continues , 2015 .

[8]  Some determinants of Bernoulli, Euler and related numbers , 1959 .

[9]  Yeong-Nan Yeh,et al.  Hankel determinants of linear combinations of consecutive Catalan-like numbers , 2017, Discret. Math..

[10]  Yifan Yang,et al.  Period polynomials and explicit formulas for Hecke operators on Γ0(2) , 2009, Mathematical Proceedings of the Cambridge Philosophical Society.

[11]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[12]  Christian Krattenthaler,et al.  The Number of Rhombus Tilings of a Symmetric Hexagon which Contain a Fixed Rhombus on the Symmetry Axis, II , 2000, Eur. J. Comb..

[13]  Johann Cigler,et al.  Some determinants of path generating functions , 2009, Adv. Appl. Math..

[14]  Some q-Orthogonal Polynomials and Related Hankel Determinants , 2002 .

[15]  Diane Y. H. Shi,et al.  Orthogonal polynomials and connection to generalized Motzkin numbers for higher-order Euler polynomials , 2019, Journal of Number Theory.

[16]  Alexandre Junod.  Hankel determinants and orthogonal polynomials , 2003 .

[17]  M. Ismail,et al.  Classical and Quantum Orthogonal Polynomials in One Variable: Bibliography , 2005 .