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- W3159224994 abstract "Abstract For each integer $$nge 1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> we consider the unique polynomials $$P, Qin {mathbb {Q}}[x]$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>P</mml:mi> <mml:mo>,</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>Q</mml:mi> <mml:mo>[</mml:mo> <mml:mi>x</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> of smallest degree n that are solutions of the equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>P</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mi>x</mml:mi> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>Q</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We derive numerous properties of these polynomials and their derivatives, including explicit expansions, differential equations, recurrence relations, generating functions, resultants, discriminants, and irreducibility results. We also consider some related polynomials and their properties." @default.
- W3159224994 created "2021-05-10" @default.
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- W3159224994 date "2021-04-29" @default.
- W3159224994 modified "2023-10-14" @default.
- W3159224994 title "Arithmetic properties of polynomial solutions of the Diophantine equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$" @default.
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- W3159224994 doi "https://doi.org/10.1007/s10998-020-00376-5" @default.
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