scholarly journals Partial sum of the products of the Horadam numbers with subscripts in arithmetic progression

2021 ◽  
Vol 27 (2) ◽  
pp. 54-63
Author(s):  
Kunle Adegoke ◽  
◽  
Robert Frontczak ◽  
Taras Goy ◽  
◽  
...  
Keyword(s):  
Arithmetic Progression ◽  
Partial Sum

We evaluate the partial sum of the products of the terms of any two Horadam sequences with subscripts in arithmetic progression. Illustrative examples are drawn from six well-known Horadam sequences.

scholarly journals Five squares in arithmetic progression over quadratic fields

2013 ◽  
Vol 29 (4) ◽  
pp. 1211-1238 ◽  
Author(s):  
Enrique González-Jiménez ◽  
Xavier Xarles
Keyword(s):  
Arithmetic Progression ◽  
Quadratic Fields

Extension of Wald's first lemma to Markov processes

1999 ◽  
Vol 36 (01) ◽  
pp. 48-59 ◽  
Author(s):  
George V. Moustakides
Keyword(s):  
Integral Equation ◽  
Markov Process ◽  
Markov Processes ◽  
Correction Term ◽  
Stopping Time ◽  
Poisson Integral ◽  
Homogeneous Markov Process ◽  
Partial Sum ◽  
Poisson Integral Equation ◽  
Scalar Nonlinearity

Let ξ0,ξ1,ξ2,… be a homogeneous Markov process and let S n denote the partial sum S n = θ(ξ1) + … + θ(ξ n ), where θ(ξ) is a scalar nonlinearity. If N is a stopping time with 𝔼N < ∞ and the Markov process satisfies certain ergodicity properties, we then show that 𝔼S N = [lim n→∞𝔼θ(ξ n )]𝔼N + 𝔼ω(ξ0) − 𝔼ω(ξ N ). The function ω(ξ) is a well defined scalar nonlinearity directly related to θ(ξ) through a Poisson integral equation, with the characteristic that ω(ξ) becomes zero in the i.i.d. case. Consequently our result constitutes an extension to Wald's first lemma for the case of Markov processes. We also show that, when 𝔼N → ∞, the correction term is negligible as compared to 𝔼N in the sense that 𝔼ω(ξ0) − 𝔼ω(ξ N ) = o(𝔼N).

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