Maji–Sarkar conjecture on zeros of higher-power Ramanujan polynomials

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Let kk and ℓ\ell be fixed natural numbers, and define

R2k+1(ℓ)(Z):=∑j=0k+1(−1)(ℓ+1)jB2jℓ((2j)!)ℓB2k+2−2jℓ((2k+2−2j)!)ℓZj.R_{2k+1}^{(\ell)}(Z):=\sum_{j=0}^{k+1}(-1)^{(\ell+1)j}\frac{B_{2j}^{\ell}}{((2j)!)^{\ell}}\frac{B_{2k+2-2j}^{\ell}}{((2k+2-2j)!)^{\ell}}Z^j.

Here BmB_m denotes the mmth Bernoulli number; the variable ZZ replaces z2ℓz^{2\ell} in the one-variable generalization of the Ramanujan polynomial. Conjecture on the zeros of R2k+1(ℓ)R_{2k+1}^{(\ell)}. All non-real roots of R2k+1(ℓ)(Z)R_{2k+1}^{(\ell)}(Z) are simple and lie on the unit circle ∣Z∣=1|Z|=1. For ℓ=1\ell=1, this polynomial coincides with the classical Ramanujan polynomial. The conjecture is motivated by numerical evidence and proposes a unit-circle property for the non-real zeros, but the supplied text gives no resolution.

References

Primary source

Bibekananda Maji and Tithi Sarkar, “Zeros of Ramanujan-type Polynomials”, arXiv:2306.10283 (2023).

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