Arithmetical Power Series Expansion of the Sigma Function for a Plane Curve

2018 ◽  
Vol 61 (4) ◽  
pp. 995-1022
Author(s):  
Yoshihiro Ônishi
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Algebraic Curve ◽  
Power Series Expansion ◽  
Plane Curve ◽  
Weierstrass Function ◽  
Sigma Function ◽  
Higher Genus ◽  
Prime 2 ◽  
Natural Way

AbstractThe Weierstrass function σ(u) associated with an elliptic curve can be generalized in a natural way to an entire function associated with a higher genus algebraic curve. This generalized multivariate sigma function has been investigated since the pioneering work of Felix Klein. The present paper shows Hurwitz integrality of the coefficients of the power series expansion around the origin of the higher genus sigma function associated with a certain plane curve, which is called an (n,s)-curve or a plane telescopic curve. For the prime (2), the expansion of the sigma function is not Hurwitz integral, but its square is. This paper clarifies the precise structure of this phenomenon. In Appendix A, computational examples for the trigonal genus 3 curve ((3, 4)-curve)y3+ (μ1x+μ4)y2+ (μ2x2+μ5x+μ8)y=x4+μ3x3+μ6x2+μ9x+μ12(whereμjare constants) are given.

scholarly journals Identities Involving the Fourth-Order Linear Recurrence Sequence

Symmetry ◽  
2019 ◽  
Vol 11 (12) ◽  
pp. 1476 ◽  
Author(s):  
Lan Qi ◽  
Zhuoyu Chen
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Fourth Order ◽  
Power Series Expansion ◽  
Linear Recurrence ◽  
Recurrence Sequence ◽  
Order Linear ◽  
Elementary Methods ◽  
Linear Recurrence Sequence

In this paper, we introduce the fourth-order linear recurrence sequence and its generating function and obtain the exact coefficient expression of the power series expansion using elementary methods and symmetric properties of the summation processes. At the same time, we establish some relations involving Tetranacci numbers and give some interesting identities.

Analytic Structure and Power-Series Expansion of the Jost Matrix

2012 ◽  
Vol 54 (5-6) ◽  
pp. 673-683
Author(s):  
S. A. Rakityansky ◽  
N. Elander
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Power Series Expansion ◽  
Analytic Structure

scholarly journals Is the Sibuya distribution a progeny?

2019 ◽  
Vol 56 (01) ◽  
pp. 52-56
Author(s):  
Gérard Letac
Keyword(s):  
Power Series ◽  
Series Expansion ◽  
Generating Function ◽  
Branching Process ◽  
Power Series Expansion ◽  
Nonnegative Coefficients ◽  
Positive Integers

AbstractFor 0 < a < 1, the Sibuya distribution sa is concentrated on the set ℕ+ of positive integers and is defined by the generating function $$\sum\nolimits_{n = 1}^\infty s_a (n)z^{{\kern 1pt} n} = 1 - (1 - z)^a$$. A distribution q on ℕ+ is called a progeny if there exists a branching process (Zn)n≥0 such that Z0 = 1, such that $$(Z_1 ) \le 1$$, and such that q is the distribution of $$\sum\nolimits_{n = 0}^\infty Z_n$$. this paper we prove that sa is a progeny if and only if $${\textstyle{1 \over 2}} \le a < 1$$. The main point is to find the values of b = 1/a such that the power series expansion of u(1 − (1 − u)b)−1 has nonnegative coefficients.

Sign in / Sign up

Export Citation Format

Share Document