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Generalized Heine–Stieltjes and Van Vleck polynomials associated with two-level, integrable BCS models

“…The correspondence between zeros of polynomials and models of physical systems has a very long history which goes back to early works of Stieltjes [1], Heine [2], Bôcher [3], Van Vleck [4] and Polya [5] at the end of the 19th century in relation to electrostatic models. A concise summary can be found in [6].…”

“…2 , α 2 = 4.9 × 10 3 , α 3 = 7.3 × 10 4 , α 4 = 8.1 × 10 6 , α 5 = 7.1 × 10 6 , α 6 = 5.1 × 107 , α 7 = 3.1 × 10 8 , α 8 = 1.6 × 10 9 , α 9 = 7.1 × 10 9 , α 10 = 2.8 × 10 10 , α 11 = 9.8 × 10 10 , α 12 = 3.1 × 10 11 , α 13 = 8.6 × 10 11 , α 14 = 2.2 × 10 12 , α 15 = 5.1 × 10 12 , α 16 = 1.1 × 10 13 , α 17 = 2.1 × 10 13 , α 18 = 3.7 × 10 13 , α 19 = 6.0 × 10 13 , α 20 = 9.1 × 10 13 , α 21 = 1.3 × 10 14 , α 22 = 1.6 × 10 14 , α 23 = 1.9 × 10 14 , α 24 = 2.1 × 10 14 , α 25 = 2.1 × 10 14 , α 26 = 2.0 × 10 14 , α 27 = 1.7 × 10 14 , α 28 = 1.4 × 10 14 , α 29 = 1.0 × 10 14 , α 30 = 7.0 × 10 13 , α 31 = 4.4 × 10 13 , α 32 = 2.6 × 10 13 , α 33 = 1.4 × 10 13 , α 34 = 6.7 × 10 12 , α 35 = 3.0 × 10 12 , α 36 = 1.2 × 10 12 , α 37 = 4.5 × 10 11 , α 38 = 1.5 × 10 11 , α 39 = 4.6 × 10 10 , α 40 = 1.2 × 10 10 , α 41 = 2.9 × 10 9 , α 42 = 6.2 × 10 8 , α 43 = 1.1 × 10 8 , α 44 = 1.8 × 10 7 , α 45 = 2.3 × 10 6 , α 46 = 2.5 × 10 5 , α 47 = 2.1 × 10 4 , α 48 = 1.3 × 10 3 , α -Stieltjes and Van Vleck polynomials associated with two-level BCS models…”