Closed-form coefficients for higher-order polygamma continued fractions

Determine explicit closed-form formulas for the coefficients of the continued-fraction representations of the polygamma functions of order k≥3, comparable to the formulas available for the trigamma and tetragamma functions.

Background

The paper derives explicit continued fractions for selected values of the Lerch transcendent and the Hurwitz zeta function, including representations connected with the trigamma function ψ₁ and tetragamma function ψ₂. These two cases admit explicit closed-form coefficient formulas through known Stieltjes-transform continued fractions. The authors state that comparable closed-form formulas are not known for the coefficients of the corresponding continued fractions for higher-order polygamma functions, leaving their determination unresolved.

References

Although explicit closed-form formulas are available for the continued fractions of the trigamma \psi_1 and tetragamma \psi_2 functions, no comparable closed-form formulas are known to us for the coefficients of the corresponding continued fractions for k\geqslant3.

— Four explicit continued fractions for values of the Lerch transcendent and the Hurwitz zeta function  (2609.08595 - Grigutis et al., 8 Sep 2026) in Section 2.3, “Prior work”