Hankel Determinants of shifted sequences of Bernoulli and Euler numbers

Abstract. Hankel determinants of sequences related to Bernoulli and Euler numbers have been studied before, and numerous identities are known. However, when a sequence is shifted by one unit, the situation often changes significantly. In this paper we use classical orthogonal polynomials and related methods to prove a general result concerning Hankel determinants for shifted sequences. We then apply this result to obtain new Hankel determinant evaluations for a total of 13 sequences related to Bernoulli and Euler numbers, one of which concerns Euler polynomials.

[1]  Guo-Niu Han.  Jacobi continued fraction and Hankel determinants of the Thue-Morse sequence , 2016 .

[2]  C. Krattenthaler.  ADVANCED DETERMINANT CALCULUS , 1999, math/9902004.

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

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

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

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

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

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

[9]  M. Anshelevich,et al.  Introduction to orthogonal polynomials , 2003 .

[10]  Karl Dilcher,et al.  Hankel determinants of sequences related to Bernoulli and Euler polynomials , 2020, International Journal of Number Theory.

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

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

[13]  J. Touchard,et al.  Nombres Exponentiels Et Nombres De Bernoulli , 1956, Canadian Journal of Mathematics.

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