2013/10/21 by Yik‐Man Chiang, Chiang, Yik-Man, Guo‐Fu Yu +1
Mathematics · #33E10 #34M05 #45B05 #Algebraic and Geometric Analysis #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Primary 30D35 #Secondary 33C47 #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1310.5507
openalex publication_date 2013/10/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Biconfluent Heun equation (BHE) is a confluent case of the general Heun equation which has one more regular singular points than the Gauss hypergeometric equation on the Riemann sphere ℂ. Motivated by a Nevanlinna theory (complex oscillation theory) approach, we have established a theory of periodic BHE (PBHE) in parallel with the Lamé equation verses the Heun equation, and the Mathieu equation verses the confluent Heun equation. We have established condition that lead to explicit construction of eigen-solutions of PBHE, and their single and double orthogonality, and a related first-order Fredholm-type integral equation for which the corresponding eigen-solutions must satisfy. We have also established a Bessel polynomials analogue at the BHE level which is based on the observation that both the Bessel equation and the BHE have a regular singular point at the origin and an irregular singular point at infinity on the Riemann sphere ℂ, and that the former equation has orthogonal polynomial solutions with respect to a complex weight. Finally, we relate our results to an equation considered by Turbiner, Bender and Dunne, etc concerning a quasi-exact solvable Schrödinger equation generated by first order operators such that the second order operators possess a finite-dimensional invariant subspace in a Lie algebra of SL2(ℂ)