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1: 24.12 Zeros
§24.12(i) Bernoulli Polynomials: Real Zeros
Let R ( n ) be the total number of real zeros of B n ( x ) . …
§24.12(ii) Euler Polynomials: Real Zeros
§24.12(iii) Complex Zeros
§24.12(iv) Multiple Zeros
2: 18.16 Zeros
§18.16(ii) Jacobi
Inequalities
§18.16(iii) Ultraspherical and Legendre
§18.16(iv) Laguerre
Asymptotic Behavior
3: 18.2 General Orthogonal Polynomials
§18.2(vi) Zeros
4: 29.20 Methods of Computation
A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree. …
§29.20(iii) Zeros
Zeros of Lamé polynomials can be computed by solving the system of equations (29.12.13) by employing Newton’s method; see §3.8(ii). …
5: 1.11 Zeros of Polynomials
§1.11 Zeros of Polynomials
Horner’s Scheme
Extended Horner Scheme
§1.11(ii) Elementary Properties
Descartes’ Rule of Signs
6: 31.15 Stieltjes Polynomials
§31.15(ii) Zeros
If z 1 , z 2 , , z n are the zeros of an n th degree Stieltjes polynomial S ( z ) , then every zero z k is either one of the parameters a j or a solution of the system of equations
31.15.2 j = 1 N γ j / 2 z k - a j + j = 1 j k n 1 z k - z j = 0 , k = 1 , 2 , , n .
See Marden (1966), Alam (1979), and Al-Rashed and Zaheer (1985) for further results on the location of the zeros of Stieltjes and Van Vleck polynomials. …
7: 3.8 Nonlinear Equations
§3.8(iv) Zeros of Polynomials
Bairstow’s Method
For further information on the computation of zeros of polynomials see McNamee (2007). …
§3.8(vi) Conditioning of Zeros
For moderate or large values of n it is not uncommon for the magnitude of the right-hand side of (3.8.14) to be very large compared with unity, signifying that the computation of zeros of polynomials is often an ill-posed problem. …
8: 28.9 Zeros
For q the zeros of ce 2 n ( z , q ) and se 2 n + 1 ( z , q ) approach asymptotically the zeros of He 2 n ( q 1 / 4 ( π - 2 z ) ) , and the zeros of ce 2 n + 1 ( z , q ) and se 2 n + 2 ( z , q ) approach asymptotically the zeros of He 2 n + 1 ( q 1 / 4 ( π - 2 z ) ) . …
9: 29.12 Definitions
In consequence they are doubly-periodic meromorphic functions of z . The superscript m on the left-hand sides of (29.12.1)–(29.12.8) agrees with the number of z -zeros of each Lamé polynomial in the interval ( 0 , K ) , while n - m is the number of z -zeros in the open line segment from K to K + i K . … The polynomial P ( ξ ) is of degree n and has m zeros (all simple) in ( 0 , 1 ) and n - m zeros (all simple) in ( 1 , k - 2 ) . … Let ξ 1 , ξ 2 , , ξ n denote the zeros of the polynomial P in (29.12.9) arranged according to …
29.12.13 ρ + 1 4 ξ p + σ + 1 4 ξ p - 1 + τ + 1 4 ξ p - k - 2 + q = 1 q p n 1 ξ p - ξ q = 0 , p = 1 , 2 , , n .
10: 18.24 Hahn Class: Asymptotic Approximations
Asymptotic approximations are also provided for the zeros of K n ( x ; p , N ) in various cases depending on the values of p and μ . … For asymptotic approximations for the zeros of M n ( n x ; β , c ) in terms of zeros of Ai ( x ) 9.9(i)), see Jin and Wong (1999) and Khwaja and Olde Daalhuis (2012). … Corresponding approximations are included for the zeros of P n ( λ ) ( n x ; ϕ ) . …