Combinatorial sums through Riordan arrays

2011 ◽  
Vol 101 (1-2) ◽  
pp. 195-210 ◽  
Author(s):  
Renzo Sprugnoli
Keyword(s):  
Riordan Arrays ◽  
Combinatorial Sums

scholarly journals Some Properties of the(p,q)-Fibonacci and(p,q)-Lucas Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
GwangYeon Lee ◽  
Mustafa Asci
Keyword(s):  
Generating Functions ◽  
Combinatorial Identities ◽  
Fibonacci Polynomials ◽  
Lucas Polynomials ◽  
Pascal Matrix ◽  
Riordan Arrays ◽  
Combinatorial Sums

Riordan arrays are useful for solving the combinatorial sums by the help of generating functions. Many theorems can be easily proved by Riordan arrays. In this paper we consider the Pascal matrix and define a new generalization of Fibonacci polynomials called(p,q)-Fibonacci polynomials. We obtain combinatorial identities and by using Riordan method we get factorizations of Pascal matrix involving(p,q)-Fibonacci polynomials.

scholarly journals Combinatorial sums and implicit Riordan arrays

2009 ◽  
Vol 309 (2) ◽  
pp. 475-486 ◽  
Author(s):  
Donatella Merlini ◽  
Renzo Sprugnoli ◽  
Maria Cecilia Verri
Keyword(s):  
Riordan Arrays ◽  
Combinatorial Sums

Summations of Inverse Generalized Harmonic Numbers with Riordan Arrays

2014 ◽  
Vol 687-691 ◽  
pp. 1394-1398
Author(s):  
Gao Wen Xi ◽  
Zheng Ping Zhang
Keyword(s):  
Stirling Numbers ◽  
Harmonic Number ◽  
Harmonic Numbers ◽  
Riordan Arrays ◽  
Combinatorial Sums ◽  
Generalized Harmonic Numbers ◽  
Riordan Array

By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we using connections between the Stirling numbers of both kinds and other inverse generalized harmonic numbers. we proved some combinatorial sums and inverse generalized harmonic number identities.

Inverse Generalized Harmonic Numbers with Riordan Arrays

2013 ◽  
Vol 842 ◽  
pp. 750-753
Author(s):  
Gao Wen Xi ◽  
Lan Long ◽  
Xue Quan Tian ◽  
Zhao Hui Chen
Keyword(s):  
Stirling Numbers ◽  
Harmonic Number ◽  
Harmonic Numbers ◽  
Riordan Arrays ◽  
Combinatorial Sums ◽  
Generalized Harmonic Numbers ◽  
Riordan Array

In this paper, By observing that the infinite triangle obtained from some generalized harmonic numbers follows a Riordan array, we obtain connections between the Stirling numbers of both kinds and other inverse generalized harmonic numbers. Further, we proved some combinatorial sums and inverse generalized harmonic number identities.

scholarly journals Riordan arrays and combinatorial sums

1994 ◽  
Vol 132 (1-3) ◽  
pp. 267-290 ◽  
Author(s):  
Renzo Sprugnoli
Keyword(s):  
Riordan Arrays ◽  
Combinatorial Sums

scholarly journals On identities involving generalized harmonic, hyperharmonic and special numbers with Riordan arrays

2021 ◽  
Vol 9 (1) ◽  
pp. 22-30
Author(s):  
Sibel Koparal ◽  
Neşe Ömür ◽  
Ömer Duran
Keyword(s):  
Stirling Numbers ◽  
Bernoulli Numbers ◽  
Riordan Arrays ◽  
Riordan Array

Abstract In this paper, by means of the summation property to the Riordan array, we derive some identities involving generalized harmonic, hyperharmonic and special numbers. For example, for n ≥ 0, ∑ k = 0 n B k k ! H ( n . k , α ) = α H ( n + 1 , 1 , α ) - H ( n , 1 , α ) , \sum\limits_{k = 0}^n {{{{B_k}} \over {k!}}H\left( {n.k,\alpha } \right) = \alpha H\left( {n + 1,1,\alpha } \right) - H\left( {n,1,\alpha } \right)} , and for n > r ≥ 0, ∑ k = r n - 1 ( - 1 ) k s ( k , r ) r ! α k k ! H n - k ( α ) = ( - 1 ) r H ( n , r , α ) , \sum\limits_{k = r}^{n - 1} {{{\left( { - 1} \right)}^k}{{s\left( {k,r} \right)r!} \over {{\alpha ^k}k!}}{H_{n - k}}\left( \alpha \right) = {{\left( { - 1} \right)}^r}H\left( {n,r,\alpha } \right)} , where Bernoulli numbers Bn and Stirling numbers of the first kind s (n, r).

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 Using Bezier curves in medical applications

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 937-943 ◽  
Author(s):  
Buket Simsek ◽  
Ahmet Yardimci
Keyword(s):  
Cross Sections ◽  
Biomedical Applications ◽  
Medical Physics ◽  
Binomial Coefficients ◽  
Medical Applications ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Combinatorial Sums ◽  
Bernstein Basis Functions

In this paper we survey the 3D reconstruction of an object from its 2D cross-sections has many applications in different fields of sciences such as medical physics and biomedical applications. The aim of this paper is to give not only the Bezier curves in medical applications, but also by using generating functions for the Bernstein basis functions and their identities, some combinatorial sums involving binomial coefficients are deriven. Finally, we give some comments related to the above areas.

scholarly journals On a Class of Combinatorial Sums Involving Generalized Factorials

2007 ◽  
Vol 2007 ◽  
pp. 1-9 ◽  
Author(s):  
W.-S. Chou ◽  
L. C. Hsu ◽  
P. J.-S. Shiue
Keyword(s):  
Stirling Numbers ◽  
Combinatorial Sums ◽  
Generalized Stirling Numbers

The object of this paper is to show that generalized Stirling numbers can be effectively used to evaluate a class of combinatorial sums involving generalized factorials.

Sign in / Sign up

Export Citation Format

Share Document