We associate recursively defined polynomials with normalized (non-vanishing) arithmetic functions g and h. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$P_0^{g,h}(x):=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>P</mi> <mn>0</mn> <mrow> <mi>g</mi> <mo>,</mo> <mi>h</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. Then, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$ P_n^{g,h}(x):= \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>P</mi> <mi>n</mi> <mrow> <mi>g</mi> <mo>,</mo> <mi>h</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>:</mo> <mo>=</mo> <mfrac> <mi>x</mi> <mrow> <mi>h</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> <msubsup> <mo>∑</mo> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mi>n</mi> </msubsup> <mi>g</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> <mspace width="0.166667em" /> <msubsup> <mi>P</mi> <mrow> <mi>n</mi> <mo>-</mo> <mi>k</mi> </mrow> <mrow> <mi>g</mi> <mo>,</mo> <mi>h</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> <mo>.</mo> </mrow> </math></EquationSource> </InlineEquation> For special functions g and h, we obtain the D’Arcais polynomials, which are associated with the coefficients of powers of the Dedekind <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\eta $</EquationSource> <EquationSource Format="MATHML"><math> <mi>η</mi> </math></EquationSource> </InlineEquation>-function and related to a hook length formula by Nekrasov and Okounkov. More examples are offered by Pochhammer polynomials, Chebyshev polynomials of the second kind, and Laguerre polynomials. We provide explicit formulas and identities for the coefficients of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$P_n^{g,h}(x)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi>P</mi> <mi>n</mi> <mrow> <mi>g</mi> <mo>,</mo> <mi>h</mi> </mrow> </msubsup> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> which separate the impact of g and h. Finally, we mention several applications.

FORMULAS FOR COEFFICIENTS OF POLYNOMIALS ASSIGNED TO ARITHMETIC FUNCTIONS

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2025年4月24日

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Bernhard Heim,
Markus Neuhauser

摘要

We associate recursively defined polynomials with normalized (non-vanishing) arithmetic functions g and h. Let $P_0^{g,h}(x):=1$ $P_{0}^{g, h} (x) : = 1$ . Then, $ P_n^{g,h}(x):= \frac{x}{h(n)} \sum _{k=1}^{n} g(k) \, P_{n-k}^{g,h}(x). $ $P_{n}^{g, h} (x) : = \frac{x}{h (n)} \sum_{k = 1}^{n} g (k) P_{n - k}^{g, h} (x) .$ For special functions g and h, we obtain the D’Arcais polynomials, which are associated with the coefficients of powers of the Dedekind $\eta $ $η$ -function and related to a hook length formula by Nekrasov and Okounkov. More examples are offered by Pochhammer polynomials, Chebyshev polynomials of the second kind, and Laguerre polynomials. We provide explicit formulas and identities for the coefficients of $P_n^{g,h}(x)$ $P_{n}^{g, h} (x)$ which separate the impact of g and h. Finally, we mention several applications.