Arithmetic properties of polynomial solutions of the Diophantine equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$
Keyword(s):
AbstractFor each integer $$n\ge 1$$ n ≥ 1 we consider the unique polynomials $$P, Q\in {\mathbb {Q}}[x]$$ P , Q ∈ Q [ x ] of smallest degree n that are solutions of the equation $$P(x)x^{n+1}+Q(x)(x+1)^{n+1}=1$$ P ( x ) x n + 1 + Q ( x ) ( x + 1 ) n + 1 = 1 . 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.
2018 ◽
Vol 4
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pp. 137-143
Keyword(s):
Keyword(s):
1985 ◽
Vol 26
(7)
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pp. 1547-1552
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1997 ◽
Vol 20
(4)
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pp. 759-768
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