Quantum Hilbert matrices and orthogonal polynomials
نویسندگان
چکیده
Using the notion of quantum integers associated with a complex number q 6= 0, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q-Jacobi polynomials when |q| < 1, and for the special value q = (1 − √ 5)/(1 + √ 5) they are closely related to Hankel matrices of reciprocal Fibonacci numbers called Filbert matrices. We find a formula for the entries of the inverse quantum Hilbert matrix. 2000 Mathematics Subject Classification: primary 33D45; secondary 11B39.
منابع مشابه
Quantum Hilbert matrices and orthogonal polynomials
Using the notion of quantum integers associated with a complex number q 6= 0, we define the quantum Hilbert matrix and various extensions. They are Hankel matrices corresponding to certain little q-Jacobi polynomials when |q| < 1, and for the special value q = (1 − √ 5)/(1 + √ 5) they are closely related to Hankel matrices of reciprocal Fibonacci numbers called Filbert matrices. We find a formu...
متن کاملA Riemann-hilbert Problem for Skew-orthogonal Polynomials
Abstract. We find a local (d + 1)× (d + 1) Riemann-Hilbert problem characterizing the skew-orthogonal polynomials associated to the partition function of the Gaussian Orthogonal Ensemble of random matrices with a potential function of degree d. Our Riemann-Hilbert problem is similar to a local d × d RiemannHilbert problem found by Kuijlaars and McLaughlin characterizing the bi-orthogonal polyno...
متن کاملFibonacci numbers and orthogonal polynomials
We prove that the sequence (1/Fn+2)n≥0 of reciprocals of the Fibonacci numbers is a moment sequence of a certain discrete probability, and we identify the orthogonal polynomials as little q-Jacobi polynomials with q = (1− √ 5)/(1+ √ 5). We prove that the corresponding kernel polynomials have integer coefficients, and from this we deduce that the inverse of the corresponding Hankel matrices (1/F...
متن کاملUniversal behavior for averages of characteristic polynomials at the origin of the spectrum
It has been shown by Strahov and Fyodorov that averages of products and ratios of characteristic polynomials corresponding to Hermitian matrices of a unitary ensemble, involve kernels related to orthogonal polynomials and their Cauchy transforms. We will show that, for the unitary ensemble 1 Ẑn | detM |2αe−nV dM of n×n Hermitian matrices, these kernels have universal behavior at the origin of t...
متن کاملNumerical solution of delay differential equations via operational matrices of hybrid of block-pulse functions and Bernstein polynomials
In this paper, we introduce hybrid of block-pulse functions and Bernstein polynomials and derive operational matrices of integration, dual, differentiation, product and delay of these hybrid functions by a general procedure that can be used for other polynomials or orthogonal functions. Then, we utilize them to solve delay differential equations and time-delay system. The method is based upon e...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Computational Applied Mathematics
دوره 233 شماره
صفحات -
تاریخ انتشار 2009