A Double Binomial Sum: Putnam 35–A4 Revisited

Abstract We give an upper estimate for the binomial sum $\begin{array}{} U_{d} \left(x\right)=\sum _{r\ge 0}\binom {d-r+1} r\cdot x^{r} \end{array} $ with natural d and real nonnegative x. In particular, this estimate implies that $\begin{array}{} U_{d} \left(x\right)=O\bigl((0.5+\sqrt{x+0.25} )^{d} \bigr) \end{array} $ with fixed x > 0 and d → ∞.

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On the Fibonacci quaternion sequence with quadruple-produce components

Acta et Commentationes Universitatis Tartuensis de Mathematica ◽

10.12697/acutm.2021.25.11 ◽

2021 ◽

Vol 25 (1) ◽

pp. 157-170

Author(s):

Orhan Diskaya ◽

Hamza Menken

Keyword(s):

Generating Functions ◽

Binomial Sum ◽

Fibonacci Quaternion ◽

Sum Formula

This paper examines the Fibonacci quaternion sequence with quadruple-produce components, and demonstrates a golden-like ratio and some identities for this sequence. Its generating and exponential generating functions are given. Along with these, its series and binomial sum formula are established.

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A binomial sum related to Wolstenholme's theorem

Journal of Number Theory ◽

10.1016/j.jnt.2009.05.010 ◽

2009 ◽

Vol 129 (11) ◽

pp. 2659-2672 ◽

Cited By ~ 2

Author(s):

Marc Chamberland ◽

Karl Dilcher

Keyword(s):

Binomial Sum

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ASYMPTOTICS OF A 3F2 POLYNOMIAL ASSOCIATED WITH THE CATALAN–LARCOMBE–FRENCH SEQUENCE

Analysis and Applications ◽

10.1142/s0219530506000802 ◽

2006 ◽

Vol 04 (04) ◽

pp. 335-344

Author(s):

NICO M. TEMME

Keyword(s):

Asymptotic Expansion ◽

Generating Function ◽

Asymptotic Method ◽

Integral Representations ◽

Binomial Sum ◽

Large N ◽

Hypergeometric Polynomial ◽

Polynomial Formula

The large n behavior of the hypergeometric polynomial [Formula: see text] is considered by using integral representations of this polynomial. This 3F2 polynomial is associated with the Catalan–Larcombe–French sequence. Several other representations are mentioned, with references to the literature, and another asymptotic method is described by using a generating function of the sequence. The results are similar to those obtained by Clark (2004) who used a binomial sum for obtaining an asymptotic expansion.

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