An Orthogonal Property of the Hypergeometric Polynomial.
نویسنده
چکیده
1. Mittag-Leffler's polynomial gn(z) has the orthogonal property fgm(-ix)gn(ix)dx/xsh(x) = m $ n m > 0-co =-2/n m=n n>O. (1.1) This is readily obtained by inverting the integral representation' co gn(ix) = (1/ir) sin (irx)f exu (tanh '/2u)ndu/sh u n > 1 (1.2)-co0 by means of Fourier's inversion formula. The resulting equation co cosech u (tanh 1/2u)n = 1/2 e-iux cosech (irx)gn(ix)dx (1.3)-OD then gives the desired relation when sh u e-iux is expanded in powers of tanh 1/2u. With the notation of the hypergeometric function the orthogonal relation may be written in the form
منابع مشابه
Modified Clebsch-gordan-type Expansions for Products of Discrete Hypergeometric Polynomials. 1
Starting from the second-order diierence hypergeometric equation satissed by the set of discrete orthogonal polynomials fp n g, we nd the analytical expressions of the expansion coeecients of any polynomial r m (x) and of the product r m (x)q j (x) in series of the set fp n g. These coeecients are given in terms of the polynomial coeecients of the second-order diierence equations satissed by th...
متن کاملRaising and lowering operators and their factorization for generalized orthogonal polynomials of hypergeometric type on homogeneous and non-homogeneous lattice
We complete the construction of raising and lowering operators, given in a previous work, for the orthogonal polynomials of hypergeometric type on nonhomogeneous lattice, and extend these operators to the generalized orthogonal polynomials, namely, those difference of orthogonal polynomials that satisfy a similar difference equation of hypergeometric type. PACS Numbers: 0210N, 0220S, 0230V, 027...
متن کاملThe Complementary Polynomials and the Rodrigues Operator of Classical Orthogonal Polynomials
From the Rodrigues representation of polynomial eigenfunctions of a second order linear hypergeometric-type differential (difference or q-difference) operator, complementary polynomials for classical orthogonal polynomials are constructed using a straightforward method. Thus a generating function in a closed form is obtained. For the complementary polynomials we present a second order linear hy...
متن کاملThe Askey Scheme for Hypergeometric Orthogonal Polynomials Viewed from Asymptotic Analysis
Many limits are known for hypergeometric orthogonal polynomials that occur in the Askey scheme. We show how asymptotic representations can be derived by using the generating functions of the polynomials. For example, we discuss the asymptotic representation of the Meixner-Pollaczek, Jacobi, Meixner, and Krawtchouk polynomials in terms of Laguerre polynomials.
متن کاملBasic Hypergeometric Functions and Orthogonal Laurent Polynomials
A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or q-hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials { 2Φ1(q−n, qb+1; q−c+b−n; q, qz)}n=0, where 0 < q < 1 and the complex parameters b, c and d are such that b = −1,−2, . . ., c− b+ 1 = −1,−2, . ...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
- Proceedings of the National Academy of Sciences of the United States of America
دوره 28 9 شماره
صفحات -
تاریخ انتشار 1942