## Higher Transcendental Functions, Volume 2 |

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Results 1-3 of 51

Page 268

10.3) have their analogues in two variables, although the

are less simple. First, we shall prove the existence of a recurrence relation,

expressing (ax + by) Pni(x, y) as a linear combination of polynomials of degree n

+ 1, ...

10.3) have their analogues in two variables, although the

**corresponding**formulasare less simple. First, we shall prove the existence of a recurrence relation,

expressing (ax + by) Pni(x, y) as a linear combination of polynomials of degree n

+ 1, ...

Page 279

which is obtained by adding the n equations (8), and is identical with the

written in the form (10) 0F,[X* + K;-X(l-rl)Al](*7' m ,! • •• m ! i n where the A-th

power of (1 ...

which is obtained by adding the n equations (8), and is identical with the

**corresponding**equation 12.5(14) for the V* , The symbolic representation may bewritten in the form (10) 0F,[X* + K;-X(l-rl)Al](*7' m ,! • •• m ! i n where the A-th

power of (1 ...

Page 377

The quartics

= (n + %) K', we have a circle, see 13. 18(1), See the figure for further details. The

function w = cn(u, k). The interior of the 13.25 377 ELLIPTIC FUNCTIONS AND ...

The quartics

**corresponding**to u2 = 0 are ovals encircling JfB. In particular, for u2= (n + %) K', we have a circle, see 13. 18(1), See the figure for further details. The

function w = cn(u, k). The interior of the 13.25 377 ELLIPTIC FUNCTIONS AND ...

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Acad addition theorem Akad algebraic Amer analytic asymptotic expansion behavior Bessel functions biorthogonal chapter coefficients complex constant contour convergence corresponding cosh defined denotes derived elliptic integrals Erdelyi expressed in terms finite Gegenbauer polynomials given harmonic polynomial hence Hermite polynomials hypergeometric functions hypergeometric series incomplete gamma functions integral representations interval Jacobi polynomials Jacobian elliptic functions Jv(z Koschmieder Laguerre polynomials Legendre Legendre's linear linearly independent London Math Meijer Neumann nomials notation obtain orthogonal polynomials orthogonal system parabolic cylinder parameter partial differential equation poles poly polynomial of degree positive power series Proc proved rational function rectangle recurrence relation right-hand side satisfy scalar product second kind sin2 solutions surface harmonics system of orthogonal Szego theory theta functions third kind Trans transformation Tricomi valid values variable vector Watson Weierstrass weight function