<p>In this paper, we address the Skolem problem for the <i>k</i>-generalized Pell numbers (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$P_n^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>P</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation>) extended to negative indices. We focus on identifying and bounding the indices <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n<0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo><</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> for which <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$P_n^{(k)}=0.$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>P</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> <mo>=</mo> <mn>0</mn> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> In particular, we establish that the zero multiplicity of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$P_n^{(k)}$</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi>P</mi> <mi>n</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msubsup> </math></EquationSource> </InlineEquation> is <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\chi _k = \lfloor k^2/4\rfloor $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>χ</mi> <mi>k</mi> </msub> <mo>=</mo> <mrow> <mo>⌊</mo> <msup> <mi>k</mi> <mn>2</mn> </msup> <mo stretchy="false">/</mo> <mn>4</mn> <mo>⌋</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> for all <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$k \in [4, 500]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>∈</mo> <mo stretchy="false">[</mo> <mn>4</mn> <mo>,</mo> <mn>500</mn> <mo stretchy="false">]</mo> </mrow> </math></EquationSource> </InlineEquation>.</p>

Solving Skolem problem for negative indexed k-generalized Pell numbers

期刊论文
发表日期：

2026年6月2日

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M. Mohapatra,
P. K. Bhoi,
G. K. Panda

摘要

In this paper, we address the Skolem problem for the k-generalized Pell numbers ( $P_n^{(k)}$ $P_{n}^{(k)}$ ) extended to negative indices. We focus on identifying and bounding the indices $n<0$ $n < 0$ for which $P_n^{(k)}=0.$ $P_{n}^{(k)} = 0 .$ In particular, we establish that the zero multiplicity of $P_n^{(k)}$ $P_{n}^{(k)}$ is $\chi _k = \lfloor k^2/4\rfloor $ $χ_{k} = ⌊ k^{2} / 4 ⌋$ for all $k \in [4, 500]$ $k \in [4, 500]$ .