空军大学学位证【言正微aptao168】assumpt

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0\leq y<x

, each quadratic polynomial is positive on the interval

(y,x)

, and

\alpha,\beta,\gamma

is a permutation of

0,a_{1}b_{2},a_{2}b_{1}

(not all 0 by assumption) such that

\alpha\leq\beta\leq\gamma

. … ►If no such branch point is accessible from the interval of integration (for example, if the integrand is

(t(3-t)(4-t))^{-3/2}

and the interval is [1,2]), then no method using this assumption succeeds. …

… ►Other assumptions are: …

… ►Ismail (2009, §2.6) discusses this in a different

N_{n}/D_{n}

notation; also note the assumption that

\mu_{0}=1

, made throughout that reference, Ismail (2009, p. 16). …

… ►If polynomials

p_{n}(x)

are generated by recurrence relation (18.2.8) under assumption of inequality (18.2.9_5) (or similarly for the other three forms) then the

p_{n}(x)

are orthogonal by Favard’s theorem, see §18.2(viii), in that the existence of a bounded non-decreasing function

\alpha(x)

(a,b)

yielding the orthogonality realtion (18.2.4_5) is guaranteed. …