scholarly journals A factorization formula for power series

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Daniel Birmajer ◽  
Juan B. Gil ◽  
Michael D. Weiner
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Inversion Formula ◽  
Constant Term ◽  
Bell Polynomials ◽  
Integer Coefficients ◽  
Factorization Formula ◽  
International Audience ◽  
Formal Power

International audience Given an odd prime p, we give an explicit factorization over the ring of formal power series with integer coefficients for certain reducible polynomials whose constant term is of the form $p^w$ with $w>1$. Our formulas are given in terms of partial Bell polynomials and rely on the inversion formula of Lagrange.

scholarly journals FACTORING POLYNOMIALS IN THE RING OF FORMAL POWER SERIES OVER ℤ

2012 ◽  
Vol 08 (07) ◽  
pp. 1763-1776 ◽  
Author(s):  
DANIEL BIRMAJER ◽  
JUAN B. GIL ◽  
MICHAEL WEINER
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Prime Power ◽  
Sufficient Conditions ◽  
Constant Term ◽  
Arbitrary Degree ◽  
Integer Coefficients ◽  
Factorization Algorithm ◽  
Factoring Polynomials ◽  
Formal Power

We consider polynomials with integer coefficients and discuss their factorization properties in ℤ[[x]], the ring of formal power series over ℤ. We treat polynomials of arbitrary degree and give sufficient conditions for their reducibility as power series. Moreover, if a polynomial is reducible over ℤ[[x]], we provide an explicit factorization algorithm. For polynomials whose constant term is a prime power, our study leads to the discussion of p-adic integers.

scholarly journals On composition of formal power series

2002 ◽  
Vol 30 (12) ◽  
pp. 761-770 ◽  
Author(s):  
Xiao-Xiong Gan ◽  
Nathaniel Knox
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Open Problem ◽  
Constant Term ◽  
Radius Of Convergence ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
Formal Power ◽  
Special Case

Given a formal power seriesg(x)=b0+b1x+b2x2+⋯and a nonunitf(x)=a1x+a2x2+⋯, it is well known that the composition ofgwithf,g(f(x)), is a formal power series. If the formal power seriesfabove is not a nonunit, that is, the constant term offis not zero, the existence of the compositiong(f(x))has been an open problem for many years. The recent development investigated the radius of convergence of a composed formal power series likefabove and obtained some very good results. This note gives a necessary and sufficient condition for the existence of the composition of some formal power series. By means of the theorems established in this note, the existence of the composition of a nonunit formal power series is a special case.

scholarly journals Bell Polynomials and Nonlinear Inverse Relations

2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Wenchang Chu
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Bell Polynomials ◽  
Expansion Formula ◽  
Connection Coefficients ◽  
Lagrange Expansion ◽  
Formal Power

By means of the Lagrange expansion formula, we establish a general pair of nonlinear inverse series relations, which are expressed via partial Bell polynomials with the connection coefficients involve an arbitrary formal power series. As applications, two examples are presented with one of them recovering the difficult theorems discovered recently by Birmajer, Gil and Weiner (2012 and 2019).

Formal power series and nearly analytic functions

1991 ◽  
Vol 57 (1) ◽  
pp. 61-70 ◽  
Author(s):  
H�kan Hedenmalm
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Analytic Functions ◽  
Formal Power

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.

On the intersection of null spaces for matrix substitutions in a non-commutative rational formal power series

2004 ◽  
Vol 339 (8) ◽  
pp. 533-538 ◽  
Author(s):  
Daniel Alpay ◽  
Dmitry S. Kalyuzhnyı̆-Verbovetzkiı̆
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Formal Power
Sign in / Sign up

Export Citation Format

Share Document