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1: 37.12 Orthogonal Polynomials on Quadratic Surfaces

§37.12 Orthogonal Polynomials on Quadratic Surfaces

►These OPs are on the quadratic surface … ►

§37.12(ii) Jacobi Polynomials on the Conic Surface

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§37.12(iv) Quadratic Transformations

… ►Then there are quadratic transformations for the polynomials (37.12.9) and (37.12.14) in terms of complex disk polynomials (37.4.11) and complex circular Hermite polynomials (37.6.3), respectively: …

2: 27.9 Quadratic Characters

§27.9 Quadratic Characters

►For an odd prime $p$, the Legendre symbol $(n|p)$ is defined as follows. …If $p$ does not divide $n$, then $(n|p)$ has the value $1$ when the quadratic congruence $x^2\equiv n(modp)$ has a solution, and the value $-1$ when this congruence has no solution. … ►If $p,q$ are distinct odd primes, then the quadratic reciprocity law states that … ►If an odd integer $P$ has prime factorization $P={\prod }_{r=1}^{\nu \left(n\right)}{p}_r^{a_r}$, then the Jacobi symbol $(n|P)$ is defined by $(n|P)={\prod }_{r=1}^{\nu \left(n\right)}{(n|p_r)}^{a_r}$, with $(n|1)=1$. …

3: 27.18 Methods of Computation: Primes

… ►An analytic approach using a contour integral of the Riemann zeta function (§25.2(i)) is discussed in Borwein et al. (2000). … ►An alternative procedure is the binary quadratic sieve of Atkin and Bernstein (Crandall and Pomerance (2005, p. 170)). …

4: 24.14 Sums

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§24.14(i) Quadratic Recurrence Relations

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§24.14(ii) Higher-Order Recurrence Relations

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5: 37.18 Orthogonal Polynomials on Quadratic Domains

§37.18 Orthogonal Polynomials on Quadratic Domains

►These OPs are on the quadratic domain …Up to an affine transformation, this includes quadratic domains bounded by the quadratic surfaces in §37.12. … ►On the quadratic domain ${𝕍}^{d+1}$, define the inner product … ►

§37.18(ii) Jacobi Polynomials on the Cone

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6: 3.8 Nonlinear Equations

… ►If $p=2$, then the convergence is quadratic; if $p=3$, then the convergence is cubic, and so on. … ►If $\zeta$ is a simple zero, then the iteration converges locally and quadratically. … ►It converges locally and quadratically for both $ℝ$ and $ℂ$. … ►The method converges locally and quadratically, except when the wanted quadratic factor is a multiple factor of $q(z)$. … ►The quadratic nature of the convergence is evident. …

7: 15.17 Mathematical Applications

… ►The logarithmic derivatives of some hypergeometric functions for which quadratic transformations exist (§15.8(iii)) are solutions of Painlevé equations. … ►Quadratic transformations give insight into the relation of elliptic integrals to the arithmetic-geometric mean (§19.22(ii)). …

8: 13.12 Products

… ►For generalizations of this quadratic relation see Majima et al. (2000). …

9: 27.22 Software

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Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below ${10}^{16}$. Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$, and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

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10: 16.6 Transformations of Variable

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Quadratic

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