scholarly journals Hankel continued fractions and Hankel determinants of the Euler numbers

2020 ◽  
Vol 373 (6) ◽  
pp. 4255-4283
Author(s):  
Guo-Niu Han
Keyword(s):  
Continued Fractions ◽  
Euler Numbers ◽  
Hankel Determinants

scholarly journals Hankel determinants of sequences related to Bernoulli and Euler polynomials

2021 ◽  
pp. 1-29
Author(s):  
Karl Dilcher ◽  
Lin Jiu
Keyword(s):  
Bernoulli Polynomials ◽  
Euler Polynomials ◽  
Hankel Determinant ◽  
Euler Numbers ◽  
Hankel Determinants ◽  
Euler Numbers And Polynomials ◽  
Dirichlet Characters ◽  
Special Cases ◽  
Generalized Bernoulli Polynomials ◽  
Derivatives Of

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.

scholarly journals On the automaticity of sequences defined by the Thue–Morse and period-doubling Stieltjes continued fractions

2020 ◽  
Vol 16 (10) ◽  
pp. 2187-2212
Author(s):  
Yining Hu ◽  
Guoniu Wei-Han
Keyword(s):  
Continued Fractions ◽  
Continued Fraction ◽  
Period Doubling ◽  
Converse Problem ◽  
Seminal Paper ◽  
Hankel Determinants ◽  
Automatic Sequence ◽  
Power Series Expansions ◽  
Partial Quotients ◽  
Continued Fraction Expansions

Continued fraction expansions of automatic numbers have been extensively studied during the last few decades. The research interests are, on one hand, in the degree or automaticity of the partial quotients following the seminal paper of Baum and Sweet in 1976, and on the other hand, in calculating the Hankel determinants and irrationality exponents, as one can find in the works of Allouche–Peyrière–Wen–Wen, Bugeaud, and the first author. This paper is motivated by the converse problem: to study Stieltjes continued fractions whose coefficients form an automatic sequence. We consider two such continued fractions defined by the Thue–Morse and period-doubling sequences, respectively, and prove that they are congruent to algebraic series in [Formula: see text] modulo [Formula: see text]. Consequently, the sequences of the coefficients of the power series expansions of the two continued fractions modulo [Formula: see text] are [Formula: see text]-automatic.

Identities related to special polynomials and combinatorial numbers

Filomat ◽  
2017 ◽  
Vol 31 (15) ◽  
pp. 4833-4844 ◽  
Author(s):  
Eda Yuluklu ◽  
Yilmaz Simsek ◽  
Takao Komatsu
Keyword(s):  
Generating Functions ◽  
Functional Equations ◽  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Euler Numbers ◽  
Special Polynomials ◽  
Central Factorial Numbers

The aim of this paper is to give some new identities and relations related to the some families of special numbers such as the Bernoulli numbers, the Euler numbers, the Stirling numbers of the first and second kinds, the central factorial numbers and also the numbers y1(n,k,?) and y2(n,k,?) which are given Simsek [31]. Our method is related to the functional equations of the generating functions and the fermionic and bosonic p-adic Volkenborn integral on Zp. Finally, we give remarks and comments on our results.

scholarly journals EULER CLASSES: SIX-FUNCTORS FORMALISM, DUALITIES, INTEGRALITY AND LINEAR SUBSPACES OF COMPLETE INTERSECTIONS

2021 ◽  
pp. 1-66
Author(s):  
Tom Bachmann ◽  
Kirsten Wickelgren
Keyword(s):  
Commutative Algebra ◽  
Vector Bundles ◽  
Euler Class ◽  
Complete Intersections ◽  
Bilinear Forms ◽  
Euler Numbers ◽  
Coherent Sheaves ◽  
Linear Subspaces ◽  
Algebraic Vector

Abstract We equate various Euler classes of algebraic vector bundles, including those of [12] and one suggested by M. J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class and give formulas for local indices at isolated zeros, both in terms of the six-functors formalism of coherent sheaves and as an explicit recipe in the commutative algebra of Scheja and Storch. As an application, we compute the Euler classes enriched in bilinear forms associated to arithmetic counts of d-planes on complete intersections in $\mathbb P^n$ in terms of topological Euler numbers over $\mathbb {R}$ and $\mathbb {C}$ .

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