Generating Functions for Formal Power Series in Noncommuting Variables

1960 ◽  
Vol 11 (6) ◽  
pp. 988
Author(s):  
K. Goldberg
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Generating Functions ◽  
Formal Power

Formal Power Series and Generating Functions

2017 ◽  
pp. 37-50
Author(s):  
C.D. Godsil
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Generating Functions ◽  
Formal Power

scholarly journals Interlacing polynomials and the veronese construction for rational formal power series

2019 ◽  
Vol 150 (1) ◽  
pp. 1-16
Author(s):  
Philip B. Zhang
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Generating Functions ◽  
Ehrhart Polynomial ◽  
Polynomial Sequence ◽  
Real Zeros ◽  
Alternative Approach ◽  
Formal Power ◽  
Interlacing Property

AbstractFixing a positive integer r and $0 \les k \les r-1$, define $f^{\langle r,k \rangle }$ for every formal power series f as $ f(x) = f^{\langle r,0 \rangle }(x^r)+xf^{\langle r,1 \rangle }(x^r)+ \cdots +x^{r-1}f^{\langle r,r-1 \rangle }(x^r).$ Jochemko recently showed that the polynomial $U^{n}_{r,k}\, h(x) := ( (1+x+\cdots +x^{r-1})^{n} h(x) )^{\langle r,k \rangle }$ has only non-positive zeros for any $r \ges \deg h(x) -k$ and any positive integer n. As a consequence, Jochemko confirmed a conjecture of Beck and Stapledon on the Ehrhart polynomial $h(x)$ of a lattice polytope of dimension n, which states that $U^{n}_{r,0}\,h(x)$ has only negative, real zeros whenever $r\ges n$. In this paper, we provide an alternative approach to Beck and Stapledon's conjecture by proving the following general result: if the polynomial sequence $( h^{\langle r,r-i \rangle }(x))_{1\les i \les r}$ is interlacing, so is $( U^{n}_{r,r-i}\, h(x) )_{1\les i \les r}$. Our result has many other interesting applications. In particular, this enables us to give a new proof of Savage and Visontai's result on the interlacing property of some refinements of the descent generating functions for coloured permutations. Besides, we derive a Carlitz identity for refined coloured permutations.

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

scholarly journals On the Real-Rootedness of the Veronese Construction for Rational Formal Power Series

2017 ◽  
Vol 2018 (15) ◽  
pp. 4780-4798 ◽  
Author(s):  
Katharina Jochemko
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
The Real ◽  
Formal Power
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