Multiple orthogonal polynomials for a Nikishin system on a star-like set A. López-García 1 1 University of South Alabama 2 A. López-García (U. South Alabama) E. Miña-Díaz 2 University of Mississippi 1 / 23 Banded Hessenberg operators • Let H be a banded Hessenberg operator with only two non-zero diagonals of the form 0 1 0 1 0 1 . . . . . .. .. .. . . .. .. . 0 H= a0 0 a1 0 a2 0 a3 .. . where an > 0 for all n ≥ 0. Number of zero diagonals in between is p ≥ 1. A. López-García (U. South Alabama) 2 / 23 Banded Hessenberg operators • Let H be a banded Hessenberg operator with only two non-zero diagonals of the form 0 1 0 1 0 1 . . . . . .. .. .. . . .. .. . 0 H= a0 0 a1 0 a2 0 a3 .. . where an > 0 for all n ≥ 0. Number of zero diagonals in between is p ≥ 1. • For Hn the principal n × n truncation of H, the monic polynomials Qn (z) = det(z In − Hn ) satisfy zQn (z) = Qn+1 (z) + an−p Qn−p (z), n ≥ p, with initial conditions Qj (z) = z j , j = 0, . . . , p. A. López-García (U. South Alabama) 2 / 23 • If n ≡ ` mod (p + 1), 0 ≤ ` ≤ p, then Qn (z) = z ` Ln (z p+1 ), deg(Ln ) = n−` , p+1 where the zeros of Ln are simple and lie on (0, ∞). A. López-García (U. South Alabama) 3 / 23 • If n ≡ ` mod (p + 1), 0 ≤ ` ≤ p, then Qn (z) = z ` Ln (z p+1 ), deg(Ln ) = n−` , p+1 where the zeros of Ln are simple and lie on (0, ∞). • Define the star S+ := {z ∈ C : z p+1 ∈ [0, ∞)}. p=2 Theorem (Eiermann–Varga 1993, He–Saff 1994, Ben Romdhane 2008) Zeros of Qn lie on S+ . A. López-García (U. South Alabama) 3 / 23 • If n ≡ ` mod (p + 1), 0 ≤ ` ≤ p, then Qn (z) = z ` Ln (z p+1 ), deg(Ln ) = n−` , p+1 where the zeros of Ln are simple and lie on (0, ∞). • Define the star S+ := {z ∈ C : z p+1 ∈ [0, ∞)}. p=2 Theorem (Eiermann–Varga 1993, He–Saff 1994, Ben Romdhane 2008) Zeros of Qn lie on S+ . The zeros of Ln and Ln−1 interlace on (0, ∞). A. López-García (U. South Alabama) 3 / 23 2 2 • Assume (an )∞ n=0 is bounded, so H : ` → ` is a bounded operator. • Resolvent functions: φj (z) = h(zI − H)−1 ~ej , ~e0 i, j = 0, . . . , p − 1, where ~ej is the jth element in the standard basis of `2 . Then Z dνj (t) φj (z) = , S+ z − t for some compactly supported complex measure νj on S+ . • (ν0 , . . . , νp−1 ) spectral measures for H. A. López-García (U. South Alabama) 4 / 23 2 2 • Assume (an )∞ n=0 is bounded, so H : ` → ` is a bounded operator. • Resolvent functions: φj (z) = h(zI − H)−1 ~ej , ~e0 i, j = 0, . . . , p − 1, where ~ej is the jth element in the standard basis of `2 . Then Z dνj (t) φj (z) = , S+ z − t for some compactly supported complex measure νj on S+ . • (ν0 , . . . , νp−1 ) spectral measures for H. Favard type theorem for Qn (Aptekarev–Kalyagin–Van Iseghem, 2000) Assume that an > 0 for all n and the numbers an are uniformly bounded. Then the polynomials Qn are multiple orthogonal with respect to the system (ν0 , . . . , νp−1 ). This means that for each j = 0, . . . , p − 1 we have Z jn − j − 1k Qn (z) z l dνj (z) = 0, l = 0, . . . , . p S+ A. López-García (U. South Alabama) 4 / 23 Theorem (Delvaux–L., 2013) Suppose that an > 0 for all n and the sequence (an ) is periodic with period r that is a multiple of p. If the numbers a0 , a1 , . . . , ar −1 satisfy r /p−1 Y n=0 r /p−1 apn > Y n=0 r /p−1 apn+1 > · · · > Y apn+p−1 , n=0 then the polynomials Qn are multi-orthogonal with respect to a Nikishin system (ν0 , . . . , νp−1 ) on S+ . A. López-García (U. South Alabama) 5 / 23 Nikishin systems on star-like sets and an inverse spectral problem • We consider an inverse spectral problem: Starting from (ν0 , . . . , νp−1 ) Nikishin system on S+ we want to analyze asymptotic behavior of an and Qn . A. López-García (U. South Alabama) 6 / 23 Nikishin systems on star-like sets and an inverse spectral problem • We consider an inverse spectral problem: Starting from (ν0 , . . . , νp−1 ) Nikishin system on S+ we want to analyze asymptotic behavior of an and Qn . • Let p ≥ 2. To construct a Nikishin system on S+ , take (Γ0 , Γ1 , . . . , Γp−1 ) a system of star-like sets given by Γj = T −1 (∆j ), j = 0, . . . , p − 1, where T (z) = z p+1 and (∆j )p−1 j=0 are compact intervals such that [0, +∞) if j even, ∆j ⊂ (−∞, 0] if j odd. We also assume that ∆j ∩ ∆j+1 = ∅ for all j = 0, . . . , p − 2. A. López-García (U. South Alabama) 6 / 23 Case p = 2 Γ1 Γ0 Nikishin system on Γ0 for p = 2: Let (σ0 , σ1 ) be a system of positive, rotationally invariant measures supported on (Γ0 , Γ1 ), respectively. Then (ν0 , ν1 ) is the Nikishin system generated by (σ0 , σ1 ) if dν0 (z) = dσ0 (z), Z dσ1 (t) dν1 (z) = dσ0 (z) = dhσ0 , σ1 i(z). Γ1 z − t A. López-García (U. South Alabama) 7 / 23 Case p = 2 Γ1 Γ0 Nikishin system on Γ0 for p = 2: Let (σ0 , σ1 ) be a system of positive, rotationally invariant measures supported on (Γ0 , Γ1 ), respectively. Then (ν0 , ν1 ) is the Nikishin system generated by (σ0 , σ1 ) if dν0 (z) = dσ0 (z), Z dσ1 (t) dν1 (z) = dσ0 (z) = dhσ0 , σ1 i(z). Γ1 z − t The measures (ν0 , ν1 ) are supported on Γ0 . A. López-García (U. South Alabama) 7 / 23 Nikishin system on Γ0 for general p: Start with positive, rotationally invariant measures (σ0 , σ1 , . . . , σp−1 ) on (Γ0 , Γ1 , . . . , Γp−1 ), respectively. A. López-García (U. South Alabama) 8 / 23 Nikishin system on Γ0 for general p: Start with positive, rotationally invariant measures (σ0 , σ1 , . . . , σp−1 ) on (Γ0 , Γ1 , . . . , Γp−1 ), respectively. Then (ν0 , ν1 , . . . , νp−1 ) is the Nikishin system generated by (σ0 , σ1 , . . . , σp−1 ) if dν0 (z) = dσ0 (z), Z dσ1 (t) dν1 (z) = dσ0 (z) = dhσ0 , σ1 i(z), Γ1 z − t Z dhσ1 , σ2 i(t) dσ0 (z) = dhσ0 , σ1 , σ2 i(z). dν2 (z) = z −t Γ1 .. . Z dνp−1 (z) = Γ1 dhσ1 , . . . , σp−1 i(t) dσ0 (z) = dhσ0 , σ1 , . . . , σp−1 i(z) z −t The measures (ν0 , . . . , νp−1 ) are all supported on Γ0 . A. López-García (U. South Alabama) 8 / 23 Hierarchy of measures on (Γ0 , . . . , Γp−1 ) • There is a hierarchy of measures sk ,j , 0 ≤ k ≤ j, 0 ≤ j ≤ p − 1 generated by the measures (σ0 , σ1 , . . . , σp−1 ). s0,0 s0,1 s1,1 s0,2 s1,2 s2,2 ··· ··· ··· .. . s0,p−1 s1,p−1 s2,p−1 .. . sp−1,p−1 • The measures sk ,j , k ≤ j ≤ p − 1 are supported on Γk and form the k th layer of the hierarchy. A. López-García (U. South Alabama) 9 / 23 Hierarchy of measures on (Γ0 , . . . , Γp−1 ) • There is a hierarchy of measures sk ,j , 0 ≤ k ≤ j, 0 ≤ j ≤ p − 1 generated by the measures (σ0 , σ1 , . . . , σp−1 ). s0,0 s0,1 s1,1 s0,2 s1,2 s2,2 ··· ··· ··· .. . s0,p−1 s1,p−1 s2,p−1 .. . sp−1,p−1 • The measures sk ,j , k ≤ j ≤ p − 1 are supported on Γk and form the k th layer of the hierarchy. • The measures sk ,j are defined inductively as follows. For k = j we set sk ,k = σk , and for 0 ≤ k < j, we set Z dsk ,j (z) = Γk +1 dsk +1,j (t) dσk (z), z −t z ∈ Γk . • The Nikishin system (ν0 , . . . , νp−1 ) is the first layer of the hierarchy: νj = s0,j , A. López-García (U. South Alabama) j = 0, . . . , p − 1. 9 / 23 Hierarchy of measures induced on (∆0 , . . . , ∆p−1 ) • The measures σj induce a hierarchy of measures µk ,j , 0 ≤ k ≤ j, 0 ≤ j ≤ p − 1 on the real line that plays a key role in the analysis. • The measures µk ,j , k ≤ j ≤ p − 1 are supported on ∆k and form the k th layer of the hierarchy. • The measures µk ,j are defined inductively as follows: For k = j we set µk ,k = σk∗ := T ∗ σk and for 0 ≤ k < j we set Z dµk ,j (t) = t ∆k +1 • b sk ,j (z) = z p+k −j µ bk ,j (z p+1 ), A. López-García (U. South Alabama) dµk +1,j (s) t −s ! dσk∗ (t), t ∈ ∆k . µ b ≡ Cauchy transform of µ. 10 / 23 Hierarchy of measures induced on (∆0 , . . . , ∆p−1 ) • The measures σj induce a hierarchy of measures µk ,j , 0 ≤ k ≤ j, 0 ≤ j ≤ p − 1 on the real line that plays a key role in the analysis. • The measures µk ,j , k ≤ j ≤ p − 1 are supported on ∆k and form the k th layer of the hierarchy. • The measures µk ,j are defined inductively as follows: For k = j we set µk ,k = σk∗ := T ∗ σk and for 0 ≤ k < j we set Z dµk ,j (t) = t ∆k +1 • b sk ,j (z) = z p+k −j µ bk ,j (z p+1 ), dµk +1,j (s) t −s ! dσk∗ (t), t ∈ ∆k . µ b ≡ Cauchy transform of µ. • For each k , the functions (1, µ bk ,k , µ bk ,k +1 , . . . , µ bk ,p−1 ) form an AT system on any interval disjoint with ∆k with respect to any multi-index with non-increasing components. A. López-García (U. South Alabama) 10 / 23 Definition of multiple orthogonal polynomials For a Nikishin system (ν0 , . . . , νp−1 ) on Γ0 , let (Qn )∞ n=0 be the sequence of monic polynomials of lowest degree that satisfy the multiple orthogonality conditions Z jn − j − 1k Qn (x) x l dνj (x) = 0, l = 0, . . . , , j = 0, . . . , p − 1. p Γ0 A. López-García (U. South Alabama) 11 / 23 Definition of multiple orthogonal polynomials For a Nikishin system (ν0 , . . . , νp−1 ) on Γ0 , let (Qn )∞ n=0 be the sequence of monic polynomials of lowest degree that satisfy the multiple orthogonality conditions Z jn − j − 1k Qn (x) x l dνj (x) = 0, l = 0, . . . , , j = 0, . . . , p − 1. p Γ0 Theorem We have 1) Qn has maximal degree n. 2) If n ≡ ` mod (p + 1), 0 ≤ ` ≤ p, then Qn (z) = z ` Pn (z p+1 ), and Pn has exactly n−` simple zeros in the interior of ∆0 . p+1 3) The zeros of Pn interlace on ∆0 . 4) The polynomials Qn satisfy the three-term recurrence relation zQn (z) = Qn+1 (z) + an Qn−p (z), j Qj (z) = z , n ≥ p, j = 0, . . . , p, and an > 0 for all n ≥ p. A. López-García (U. South Alabama) 11 / 23 The polynomials Pn in Qn (z) = z ` Pn (z p+1 ), n≡` mod (p + 1) satisfy multi-orthogonality conditions with respect to the measures µ0,j : For each j = 0, . . . , p − 1, Z n + p` − 1 − j(p + 1) `−j Pn (t) t s dµ0,j (t) = 0, ≤s≤ . p+1 p(p + 1) ∆0 A. López-García (U. South Alabama) 12 / 23 Second kind functions and associated polunomials • Define functions Ψn,k , k = 0, . . . , p − 1 inductively (on k ) by Ψn,0 := Qn , Z Ψn,k −1 (t) Ψn,k (z) = dσk −1 (t), z −t Γk −1 • Ψn,k ∈ H(C \ Γk −1 ), k = 1, . . . , p − 1. k = 1, . . . , p − 1. • Ψn,k satisfies orthogonality conditions with respect to the measures sk ,j , k ≤ j ≤ p − 1 in the k th layer of the Nikishin hierarchy. For each j = k , . . . , p − 1, Z n−j −1 . Ψn,k (z) z l dsk ,j (z) = 0, 0≤l≤ p Γk A. López-García (U. South Alabama) 13 / 23 • Assume n ≡ ` mod (p + 1), 0 ≤ ` ≤ p. For every k = 0, . . . , p − 1, z k −` Ψn,k (z) = φn,k (z p+1 ), where φn,k ∈ H(C \ ∆k −1 ). A. López-García (U. South Alabama) 14 / 23 • Assume n ≡ ` mod (p + 1), 0 ≤ ` ≤ p. For every k = 0, . . . , p − 1, z k −` Ψn,k (z) = φn,k (z p+1 ), where φn,k ∈ H(C \ ∆k −1 ). • The functions φn,k are "almost" second-kind functions: φn,0 (z) = Pn (z), R ∆k −1 φn,k (z) = R z ∆ t φn,k −1 (t) z−t k −1 A. López-García (U. South Alabama) dσk∗−1 (t), φn,k −1 (t) z−t if 1 ≤ k ≤ `, dσk∗−1 (t), if ` < k ≤ p − 1. 14 / 23 • Assume n ≡ ` mod (p + 1), 0 ≤ ` ≤ p. For every k = 0, . . . , p − 1, z k −` Ψn,k (z) = φn,k (z p+1 ), where φn,k ∈ H(C \ ∆k −1 ). • The functions φn,k are "almost" second-kind functions: φn,0 (z) = Pn (z), R ∆k −1 φn,k (z) = R z ∆ t φn,k −1 (t) z−t k −1 dσk∗−1 (t), φn,k −1 (t) z−t if 1 ≤ k ≤ `, dσk∗−1 (t), if ` < k ≤ p − 1. • For each k , the function φn,k satisfy multi-orthogonality conditions with respect to the measures µk ,j , k ≤ j ≤ p − 1 in the k th layer of the hierarchy on the real line. A. López-García (U. South Alabama) 14 / 23 Counting the number of zeros of the functions φn,k Proposition Let 1 ≤ k ≤ p − 1, and suppose that n = λp(p + 1) + r , Then φn,k (z) = O(z A. López-García (U. South Alabama) −N(n,k ) 0 ≤ r ≤ p(p + 1) − 1. ), r −k +1 r N(n, k ) = λ + − . p p+1 15 / 23 Counting the number of zeros of the functions φn,k Proposition Let 1 ≤ k ≤ p − 1, and suppose that n = λp(p + 1) + r , Then φn,k (z) = O(z −N(n,k ) 0 ≤ r ≤ p(p + 1) − 1. ), r −k +1 r N(n, k ) = λ + − . p p+1 Proposition Let 1 ≤ k ≤ p − 1. Then all the zeros of φn,k in C \ (∆k −1 ∪ {0}) are simple and lie on the interior of ∆k . The zeros of φn+1,k and φn,k interlace. If Z (n, k ) is the number of zeros of φn,k in the interior of ∆k , then Z (n, k ) can be computed inductively as follows: Set Z (n, 0) = n−` , p+1 n≡` mod (p + 1), and for k ≥ 1, Z (n, k ) = Z (n, k − 1) − N(n, k ), if ` < k , Z (n, k − 1) + 1 − N(n, k ), if k ≤ `. A. López-García (U. South Alabama) 15 / 23 Polynomials Pn,k associated with the functions φn,k For 0 ≤ k ≤ p − 1, let Pn,k denote the monic polynomial whose zeros are the zeros of φn,k in the interior of ∆k . Proposition The function φn,k satisfy the following orthogonality conditions with respect to varying measures. If k ≥ `, then Z φn,k (t) t l dσk∗ (t) = 0, Pn,k +1 (t) l = 0, . . . , Z (n, k ) − 1, φn,k (t) t l t dσk∗ (t) = 0, Pn,k +1 (t) l = 0, . . . , Z (n, k ) − 1. ∆k and if k < `, then Z ∆k A. López-García (U. South Alabama) 16 / 23 Ratio asymptotics • We consider the p families ∞ Pn+1,k , Pn,k n=0 A. López-García (U. South Alabama) k = 0, . . . , p − 1. 17 / 23 Ratio asymptotics • We consider the p families ∞ Pn+1,k , Pn,k n=0 k = 0, . . . , p − 1. • For every k = 0, . . . , p − 1 fixed, deg(Pn+1,k ) − deg(Pn,k ) = Z (n + 1, k ) − Z (n, k ) is a periodic function of n with period p(p + 1). Moreover, Z (n + 1, k ) − Z (n, k ) ∈ {−1, 0, 1}, A. López-García (U. South Alabama) for all n. 17 / 23 Using a method devised by Aptekarev–López Lagomasino–Rocha for Nikishin systems on the real line, we can prove: Theorem Let (ν0 , ν1 , . . . , νp−1 ) be a Nikishin system generated by the positive, rotationally symmetric measures (σ0 , σ1 , . . . , σp−1 ) on the compact star-like sets (Γ0 , Γ1 , . . . , Γp−1 ). Suppose that for all j = 0, . . . , p − 1, the measure σj satisfies A. López-García (U. South Alabama) dσj (x) |dx| > 0 for a.e. x ∈ Γj . 18 / 23 Using a method devised by Aptekarev–López Lagomasino–Rocha for Nikishin systems on the real line, we can prove: Theorem Let (ν0 , ν1 , . . . , νp−1 ) be a Nikishin system generated by the positive, rotationally symmetric measures (σ0 , σ1 , . . . , σp−1 ) on the compact star-like sets (Γ0 , Γ1 , . . . , Γp−1 ). Suppose that for all j = 0, . . . , p − 1, the measure σj satisfies dσj (x) |dx| > 0 for a.e. x ∈ Γj . Then for any fixed r = 0, . . . , p(p + 1) − 1 and any fixed k = 0, . . . , p − 1, the following limit holds Pmp(p+1)+r +1,k (z) (r ) lim = Fk (z), z ∈ C \ ∆k . m→∞ Pmp(p+1)+r ,k (z) (r ) The functions Fk are independent of the measures (σ0 , . . . , σp−1 ). They only depend on the intervals (∆0 , . . . , ∆p−1 ). A. López-García (U. South Alabama) 18 / 23 Using a method devised by Aptekarev–López Lagomasino–Rocha for Nikishin systems on the real line, we can prove: Theorem Let (ν0 , ν1 , . . . , νp−1 ) be a Nikishin system generated by the positive, rotationally symmetric measures (σ0 , σ1 , . . . , σp−1 ) on the compact star-like sets (Γ0 , Γ1 , . . . , Γp−1 ). Suppose that for all j = 0, . . . , p − 1, the measure σj satisfies dσj (x) |dx| > 0 for a.e. x ∈ Γj . Then for any fixed r = 0, . . . , p(p + 1) − 1 and any fixed k = 0, . . . , p − 1, the following limit holds Pmp(p+1)+r +1,k (z) (r ) lim = Fk (z), z ∈ C \ ∆k . m→∞ Pmp(p+1)+r ,k (z) (r ) The functions Fk are independent of the measures (σ0 , . . . , σp−1 ). They only depend on the intervals (∆0 , . . . , ∆p−1 ). This result complements results by Rakhmanov (1977) Aptekarev–López Lagomasino–Rocha (2005) L. (2010) A. López-García (U. South Alabama) 18 / 23 Consequently, we obtain: 1) Ratio asymptotics for the original polynomials Qn : For any fixed r = 0, . . . , p(p + 1) − 1, lim Qmp(p+1)+r +1 (z) (r ) = z F0 (z p+1 ), Qmp(p+1)+r (z) lim Qmp(p+1)+r +1 (z) F (z p+1 ) , = 0 p Qmp(p+1)+r (z) z m→∞ z ∈ C \ Γ0 , if r 6≡ p mod (p + 1), (r ) m→∞ A. López-García (U. South Alabama) z ∈ C \ (Γ0 ∪ {0}), if r ≡ p mod (p + 1). 19 / 23 Consequently, we obtain: 1) Ratio asymptotics for the original polynomials Qn : For any fixed r = 0, . . . , p(p + 1) − 1, lim Qmp(p+1)+r +1 (z) (r ) = z F0 (z p+1 ), Qmp(p+1)+r (z) lim Qmp(p+1)+r +1 (z) F (z p+1 ) , = 0 p Qmp(p+1)+r (z) z m→∞ z ∈ C \ Γ0 , if r 6≡ p mod (p + 1), (r ) m→∞ z ∈ C \ (Γ0 ∪ {0}), if r ≡ p mod (p + 1). 2) Asymptotics for the recurrence coefficients an : For any fixed r = 0, . . . , p(p + 1) − 1, lim amp(p+1)+r = a(r ) > 0. m→∞ A. López-García (U. South Alabama) 19 / 23 Consequently, we obtain: 1) Ratio asymptotics for the original polynomials Qn : For any fixed r = 0, . . . , p(p + 1) − 1, lim Qmp(p+1)+r +1 (z) (r ) = z F0 (z p+1 ), Qmp(p+1)+r (z) lim Qmp(p+1)+r +1 (z) F (z p+1 ) , = 0 p Qmp(p+1)+r (z) z m→∞ z ∈ C \ Γ0 , if r 6≡ p mod (p + 1), (r ) m→∞ z ∈ C \ (Γ0 ∪ {0}), if r ≡ p mod (p + 1). 2) Asymptotics for the recurrence coefficients an : For any fixed r = 0, . . . , p(p + 1) − 1, lim amp(p+1)+r = a(r ) > 0. m→∞ 3) As z → ∞, (r ) F0 (z) = 1 − (r ) a(r ) +O z F0 (z) = z − a(r ) + O A. López-García (U. South Alabama) 1 z2 1 , z , if r 6≡ p mod (p + 1), if r ≡ p mod (p + 1). 19 / 23 (r ) Relations between the functions Fk and the values a(r ) • For every λ = 0, . . . , p − 1, (λ(p+1)+p) F0 A. López-García (U. South Alabama) (z) = z F (λ(p+1)) (z) 20 / 23 (r ) Relations between the functions Fk and the values a(r ) • For every λ = 0, . . . , p − 1, (λ(p+1)+p) F0 (z) = z F (λ(p+1)) (z) • So for every λ = 0, . . . , p − 1, a(λ(p+1)+p) = a(λ(p+1)) . A. López-García (U. South Alabama) 20 / 23 (r ) Relations between the functions Fk and the values a(r ) • For every λ = 0, . . . , p − 1, (λ(p+1)+p) F0 (z) = z F (λ(p+1)) (z) • So for every λ = 0, . . . , p − 1, a(λ(p+1)+p) = a(λ(p+1)) . • From the three-term recurrence relation zQn = Qn+1 + an Qn−p , we get r −1 Y (r ) (i) F0 (z), a(r ) = 1 − F0 (z) r 6≡ p mod (p + 1), i=r −p r −1 Y (r ) (i) F0 (z), a(r ) = z − F0 (z) r ≡ p mod (p + 1). i=r −p A. López-García (U. South Alabama) 20 / 23 • The previous relations imply that m0 (p+1)+p−1 m(p+1)+p−1 Y (i) F0 (z) i=m(p+1) A. López-García (U. South Alabama) = Y (i) F0 (z), 0 ≤ m, m0 ≤ p − 1, m 6= m0 . i=m0 (p+1) 21 / 23 • The previous relations imply that m0 (p+1)+p−1 m(p+1)+p−1 Y (i) F0 (z) = (i) F0 (z), 0 ≤ m, m0 ≤ p − 1, m 6= m0 . i=m0 (p+1) i=m(p+1) m0 (p+1)+p−1 m(p+1)+p−1 X Y a(r ) = r =m(p+1) A. López-García (U. South Alabama) X a(r ) , 0 ≤ m, m0 ≤ p − 1, m 6= m0 . r =m0 (p+1) 21 / 23 • The previous relations imply that m0 (p+1)+p−1 m(p+1)+p−1 Y (i) F0 (z) = (i) F0 (z), 0 ≤ m, m0 ≤ p − 1, m 6= m0 . i=m0 (p+1) i=m(p+1) m0 (p+1)+p−1 m(p+1)+p−1 X Y a(r ) = X a(r ) , 0 ≤ m, m0 ≤ p − 1, m 6= m0 . r =m0 (p+1) r =m(p+1) • We also have 0 (m0 (p+1)) 1 − F0 (z) a(m (p+1)) = , (m(p+1)) a(m(p+1)) 1 − F0 (z) z ∈ C \ ∆0 , with m, m0 as before. A. López-García (U. South Alabama) 21 / 23 Main ideas in the proof of ratio asymptotics 1) For every k = 0, . . . , p − 1, deg(Pn+1,k ) − deg(Pn,k ) = Z (n + 1, k ) − Z (n, k ) is periodic in n with period p(p + 1). A. López-García (U. South Alabama) 22 / 23 Main ideas in the proof of ratio asymptotics 1) For every k = 0, . . . , p − 1, deg(Pn+1,k ) − deg(Pn,k ) = Z (n + 1, k ) − Z (n, k ) is periodic in n with period p(p + 1). 2) The zeros of Pn+1,k and Pn,k interlace, hence for every fixed r = 0, . . . , p(p + 1) − 1, the families Pmp(p+1)+r +1,k ∞ , k = 0, . . . , p − 1 Pmp(p+1)+r ,k m=0 are normal in C \ ∆k , respectively. A. López-García (U. South Alabama) 22 / 23 Main ideas in the proof of ratio asymptotics 1) For every k = 0, . . . , p − 1, deg(Pn+1,k ) − deg(Pn,k ) = Z (n + 1, k ) − Z (n, k ) is periodic in n with period p(p + 1). 2) The zeros of Pn+1,k and Pn,k interlace, hence for every fixed r = 0, . . . , p(p + 1) − 1, the families Pmp(p+1)+r +1,k ∞ , k = 0, . . . , p − 1 Pmp(p+1)+r ,k m=0 are normal in C \ ∆k , respectively. 3) There exists Λ ⊂ N such that lim m∈Λ Pmp(p+1)+r +1,k (z) (r ) = Fk (z), Pmp(p+1)+r ,k (z) C \ ∆k , for every k = 0, . . . , p − 1. (r ) The goal is to show that the limits Fk are independent of the subsequence Λ. A. López-García (U. South Alabama) 22 / 23 4) Using the orthogonalities of φn,k with respect to varying measures and results by de la Calle – López Lagomasino on relative and ratio asymptotics of OPRL with respect to varying measures, we can show (r ) Fk (r ) (r ) is a Szegő function with weights involving Fk −1 and Fk +1 , for every k = 0, . . . , p − 1. A. López-García (U. South Alabama) 23 / 23 4) Using the orthogonalities of φn,k with respect to varying measures and results by de la Calle – López Lagomasino on relative and ratio asymptotics of OPRL with respect to varying measures, we can show (r ) Fk (r ) (r ) is a Szegő function with weights involving Fk −1 and Fk +1 , for every k = 0, . . . , p − 1. 5) Using the boundary value behavior of Szegő functions, we obtain a system of (r ) boundary value equations for the functions Fk , k = 0, . . . , p − 1. Using a result of Aptekarev on systems of boundary value equations for harmonic functions, we (r ) show that functions Fk are unique. A. López-García (U. South Alabama) 23 / 23