scholarly journals Sums and Partial Sums of Horadam Sequences: The Sum Formulas of $\sum_{k=0}^{n}x^{k}W_{k}$ and $ \sum_{k=n}^{n+m}x^{k}W_{k}$ via Generating Functions

2021 ◽  
Author(s):  
Yüksel Soykan
Keyword(s):  
Recent Result ◽  
Generating Functions ◽  
Partial Sums

In this paper, we present sums and partial sums of Horadam sequences via generating functions which extends a recent result of Prodinger.

scholarly journals Basic Properties of the Polygonal Sequence

2021 ◽  
Author(s):  
Kunle Adegoke
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Partial Sums ◽  
Basic Properties ◽  
Polygonal Numbers

We study various properties of the polygonal numbers; such as their recurrence relations, fundamental identities, weighted binomial and ordinary sums and the partial sums and generating functions of their powers. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers as is the case in most literature.

scholarly journals Basic Properties of the Polygonal Sequence

2021 ◽  
Author(s):  
Kunle Adegoke
Keyword(s):  
Generating Functions ◽  
Continued Fraction ◽  
Recurrence Relations ◽  
Partial Sums ◽  
Basic Properties ◽  
Polygonal Numbers

We study various properties of the polygonal numbers; such as their recurrence relations; fundamental identities; weighted binomial and ordinary sums; partial sums and generating functions of their powers; and a continued fraction representation for them. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers; unlike what obtains in most literature.

scholarly journals Distribution of Crossings, Nestings and Alignments of Two Edges in Matchings and Partitions

10.37236/1059 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Anisse Kasraoui ◽  
Jiang Zeng
Keyword(s):  
Recent Result ◽  
Generating Functions ◽  
Continued Fraction ◽  
Perfect Matchings ◽  
Symmetric Distribution ◽  
Set Partitions ◽  
Continued Fraction Expansions

We construct an involution on set partitions which keeps track of the numbers of crossings, nestings and alignments of two edges. We derive then the symmetric distribution of the numbers of crossings and nestings in partitions, which generalizes a recent result of Klazar and Noy in perfect matchings. By factorizing our involution through bijections between set partitions and some path diagrams we obtain the continued fraction expansions of the corresponding ordinary generating functions.

scholarly journals Summing a Family of Generalized Pell Numbers

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Linear Combination ◽  
Computer Algebra ◽  
Generating Functions ◽  
Fibonacci Numbers ◽  
Partial Sums ◽  
Pell Numbers ◽  
New Family ◽  
Binet Formula

AbstractA new family of generalized Pell numbers was recently introduced and studied by Bród ([2]). These numbers possess, as Fibonacci numbers, a Binet formula. Using this, partial sums of arbitrary powers of generalized Pell numbers can be summed explicitly. For this, as a first step, a power P lnis expressed as a linear combination of Pmn. The summation of such expressions is then manageable using generating functions. Since the new family contains a parameter R = 2r, the relevant manipulations are quite involved, and computer algebra produced huge expressions that where not trivial to handle at times.

scholarly journals On sums of coefficients of polynomials related to the Borwein conjectures

2021 ◽  
Author(s):  
Ankush Goswami ◽  
Venkata Raghu Tej Pantangi
Keyword(s):  
Number Theory ◽  
Lower Bound ◽  
Asymptotic Formula ◽  
Recent Result ◽  
Asymptotic Estimate ◽  
Partial Sums ◽  
Absolute Value ◽  
Partial Sum ◽  
Special Cases ◽  
Error Terms

AbstractRecently, Li (Int J Number Theory, 2020) obtained an asymptotic formula for a certain partial sum involving coefficients for the polynomial in the First Borwein conjecture. As a consequence, he showed the positivity of this sum. His result was based on a sieving principle discovered by himself and Wan (Sci China Math, 2010). In fact, Li points out in his paper that his method can be generalized to prove an asymptotic formula for a general partial sum involving coefficients for any prime $$p>3$$ p > 3 . In this work, we extend Li’s method to obtain asymptotic formula for several partial sums of coefficients of a very general polynomial. We find that in the special cases $$p=3, 5$$ p = 3 , 5 , the signs of these sums are consistent with the three famous Borwein conjectures. Similar sums have been studied earlier by Zaharescu (Ramanujan J, 2006) using a completely different method. We also improve on the error terms in the asymptotic formula for Li and Zaharescu. Using a recent result of Borwein (JNT 1993), we also obtain an asymptotic estimate for the maximum of the absolute value of these coefficients for primes $$p=2, 3, 5, 7, 11, 13$$ p = 2 , 3 , 5 , 7 , 11 , 13 and for $$p>15$$ p > 15 , we obtain a lower bound on the maximum absolute value of these coefficients for sufficiently large n.

scholarly journals Basic Properties of the Polygonal Sequence

2021 ◽  
Author(s):  
Kunle Adegoke
Keyword(s):  
Generating Functions ◽  
Recurrence Relations ◽  
Partial Sums ◽  
Basic Properties ◽  
Polygonal Numbers

We study various properties of the polygonal numbers; such as their recurrence relations, fundamental identities, weighted binomial and ordinary sums and the partial sums and generating functions of their powers. A feature of our results is that they are presented naturally in terms of the polygonal numbers themselves and not in terms of arbitrary integers as is the case in most literature.

Turán type inequalities for the partial sums of the generating functions of Bernoulli and Euler numbers

2012 ◽  
Vol 285 (17-18) ◽  
pp. 2129-2156 ◽  
Author(s):  
Stamatis Koumandos ◽  
Henrik Laurberg Pedersen
Keyword(s):  
Generating Functions ◽  
Partial Sums ◽  
Euler Numbers ◽  
Turán Type Inequalities

Short Generating Functions for some Semigroup Algebras

10.37236/1729 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Graham Denham
Keyword(s):  
Rational Function ◽  
Generating Functions ◽  
Hilbert Series ◽  
Simple Expression ◽  
Arbitrary Field ◽  
Graded Algebra ◽  
Semigroup Algebras ◽  
Positive Integers

Let $a_1,\ldots,a_n$ be distinct, positive integers with $(a_1,\ldots,a_n)=1$, and let k be an arbitrary field. Let $H(a_1,\ldots,a_n;z)$ denote the Hilbert series of the graded algebra k$[t^{a_1},t^{a_2},\ldots,t^{a_n}]$. We show that, when $n=3$, this rational function has a simple expression in terms of $a_1,a_2,a_3$; in particular, the numerator has at most six terms. By way of contrast, it is known that no such expression exists for any $n\geq4$.

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