2-Adic valuations of coefficients of certain integer powers of formal power series

2019 ◽  
Vol 15 (04) ◽  
pp. 723-762
Author(s):  
Maciej Ulas
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Generating Function ◽  
Integer Sequence ◽  
Formal Power

Let [Formula: see text] be an integer sequence and [Formula: see text] be its ordinary generating function. In this paper, we study the behavior of 2-adic valuations of the sequence [Formula: see text], where [Formula: see text] is fixed and [Formula: see text] More precisely, we propose a method, which under suitable assumptions on the sequence [Formula: see text] allows us to prove boundedness of the sequence [Formula: see text] for certain values of [Formula: see text]. In particular, if [Formula: see text] is the classical Rudin–Shapiro sequence, then we prove that [Formula: see text] for given [Formula: see text] and all [Formula: see text]. A similar result is proved for a relative of the Rudin–Shapiro sequence recently introduced by Lafrance, Rampersad and Yee.

scholarly journals Affine screening operators, affine Laumon spaces and conjectures concerning non-stationary Ruijsenaars functions

2019 ◽  
Vol 4 (1) ◽  
Author(s):  
Jun’ichi Shiraishi
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Generating Function ◽  
Vertex Operators ◽  
Poincaré Duality ◽  
Euler Characteristics ◽  
Poincare Duality ◽  
Main Conjecture ◽  
Formal Power

Abstract Based on the screened vertex operators associated with the affine screening operators, we introduce the formal power series $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,t)$ which we call the non-stationary Ruijsenaars function. We identify it with the generating function for the Euler characteristics of the affine Laumon spaces. When the parameters $s$ and $\kappa$ are suitably chosen, the limit $t\rightarrow q$ of $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,q/t)$ gives us the dominant integrable characters of $\widehat{\mathfrak sl}_N$ multiplied by $1/(p^N;p^N)_\infty$ (i.e. the $\widehat{\mathfrak gl}_1$ character). Several conjectures are presented for $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,t)$, including the bispectral and the Poincaré dualities, and the evaluation formula. The main conjecture asserts that (i) one can normalize $f^{\widehat{\mathfrak gl}_N}(x,p|s,\kappa|q,t)$ in such a way that the limit $\kappa\rightarrow 1$ exists, and (ii) the limit $f^{{\rm st.}\,\widehat{\mathfrak gl}_N}(x,p|s|q,t)$ gives us the eigenfunction of the elliptic Ruijsenaars operator. The non-stationary affine $q$-difference Toda operator ${\mathcal T}^{\widehat{\mathfrak gl}_N}(\kappa)$ is introduced, which comes as an outcome of the study of the Poincaré duality conjecture in the affine Toda limit $t\rightarrow 0$. The main conjecture is examined also in the limiting cases of the affine $q$-difference Toda ($t\rightarrow 0$), and the elliptic Calogero–Sutherland ($q,t\rightarrow 1$) equations.

scholarly journals Stokes factors and multilogarithms

2012 ◽  
Vol 2013 (682) ◽  
pp. 89-128
Author(s):  
Tom Bridgeland ◽  
Valerio Toledano Laredo
Keyword(s):  
Power Series ◽  
Algebraic Group ◽  
Formal Power Series ◽  
Generating Function ◽  
Lie Series ◽  
Abelian Categories ◽  
Meromorphic Connection ◽  
Formal Power

Abstract. Let G be a complex, affine algebraic group and a meromorphic connection on the trivial G-bundle over , with a pole of order 2 at zero and a pole of order 1 at infinity. We show that the map taking the residue of at zero to the corresponding Stokes factors is given by an explicit, universal Lie series whose coefficients are multilogarithms. Using a non-commutative analogue of the compositional inversion of formal power series, we show that the same holds for the inverse of , and that the corresponding Lie series coincides with the generating function for counting invariants in abelian categories constructed by D. Joyce.

scholarly journals On transformations of formal power series

2003 ◽  
Vol 184 (2) ◽  
pp. 369-383 ◽  
Author(s):  
Manfred Droste ◽  
Guo-Qiang Zhang
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power

scholarly journals On the Square Root of a Bell Matrix

2021 ◽  
Vol 76 (1) ◽  
Author(s):  
Donatella Merlini
Keyword(s):  
Power Series ◽  
Closed Form ◽  
Formal Power Series ◽  
Computational Method ◽  
Square Root ◽  
Riordan Arrays ◽  
Formal Power

AbstractIn the context of Riordan arrays, the problem of determining the square root of a Bell matrix $$R={\mathcal {R}}(f(t)/t,\ f(t))$$ R = R ( f ( t ) / t , f ( t ) ) defined by a formal power series $$f(t)=\sum _{k \ge 0}f_kt^k$$ f ( t ) = ∑ k ≥ 0 f k t k with $$f(0)=f_0=0$$ f ( 0 ) = f 0 = 0 is presented. It is proved that if $$f^\prime (0)=1$$ f ′ ( 0 ) = 1 and $$f^{\prime \prime }(0)\ne 0$$ f ″ ( 0 ) ≠ 0 then there exists another Bell matrix $$H={\mathcal {R}}(h(t)/t,\ h(t))$$ H = R ( h ( t ) / t , h ( t ) ) such that $$H*H=R;$$ H ∗ H = R ; in particular, function h(t) is univocally determined by a symbolic computational method which in many situations allows to find the function in closed form. Moreover, it is shown that function h(t) is related to the solution of Schröder’s equation. We also compute a Riordan involution related to this kind of matrices.

scholarly journals Two Interacting Coordinate Hopf Algebras of Affine Groups of Formal Series on a Category

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Laurent Poinsot
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Semidirect Product ◽  
Hopf Algebras ◽  
Formal Series ◽  
Product Structure ◽  
Locally Finite ◽  
Commutative Algebras ◽  
Affine Groups ◽  
Formal Power

A locally finite category is defined as a category in which every arrow admits only finitely many different ways to be factorized by composable arrows. The large algebra of such categories over some fields may be defined, and with it a group of invertible series (under multiplication). For certain particular locally finite categories, a substitution operation, generalizing the usual substitution of formal power series, may be defined, and with it a group of reversible series (invertible under substitution). Moreover, both groups are actually affine groups. In this contribution, we introduce their coordinate Hopf algebras which are both free as commutative algebras. The semidirect product structure obtained from the action of reversible series on invertible series by anti-automorphisms gives rise to an interaction at the level of their coordinate Hopf algebras under the form of a smash coproduct.

Cyclic codes over formal power series rings

2011 ◽  
Vol 31 (1) ◽  
pp. 331-343 ◽  
Author(s):  
Steven T. Dougherty ◽  
Liu Hongwei
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Cyclic Codes ◽  
Power Series Rings ◽  
Formal Power
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