Mathematics
http://repository.iitgn.ac.in/handle/123456789/547
Tue, 27 Jun 2017 03:37:06 GMT2017-06-27T03:37:06ZA generalized modified Bessel function and a higher level analogue of the theta transformation formula
http://repository.iitgn.ac.in/handle/123456789/2992
A generalized modified Bessel function and a higher level analogue of the theta transformation formula
Dixit, Atul; Kesarwani, Aashita; Moll, Victor H.; Temme, Nico M.
Thu, 01 Jun 2017 00:00:00 GMThttp://repository.iitgn.ac.in/handle/123456789/29922017-06-01T00:00:00ZEigenvalue problem for fractional Kirchhoff Laplacian
http://repository.iitgn.ac.in/handle/123456789/2986
Eigenvalue problem for fractional Kirchhoff Laplacian
Tyagi, Jagmohan
Sun, 01 Jan 2017 00:00:00 GMThttp://repository.iitgn.ac.in/handle/123456789/29862017-01-01T00:00:00ZPositive solution of extremal Pucci’s equations with singular and sublinear nonlinearity
http://repository.iitgn.ac.in/handle/123456789/2985
Positive solution of extremal Pucci’s equations with singular and sublinear nonlinearity
Tyagi, Jagmohan; Verma, Ram Baran
In this paper, we establish the existence of a positive solution to
{−M+λ,Λ(D2u)=μk(x)f(u)uα−ηh(x)uqu=0in Ωon ∂Ω,
{−Mλ,Λ+(D2u)=μk(x)f(u)uα−ηh(x)uqin Ωu=0on ∂Ω,
where ΩΩ is a smooth bounded domain in Rn, n≥1.Rn, n≥1. Under certain conditions on k,f and h,k,f and h, using viscosity sub- and super solution method with the aid of comparison principle, we establish the existence of a unique positive viscosity solution. This work extends and complements the earlier works on semilinear and singular elliptic equations with sublinear nonlinearity.
Thu, 01 Jun 2017 00:00:00 GMThttp://repository.iitgn.ac.in/handle/123456789/29852017-06-01T00:00:00ZAsymptotics and exact formulas for Zagier polynomials
http://repository.iitgn.ac.in/handle/123456789/2981
Asymptotics and exact formulas for Zagier polynomials
Dixit, Atul; Glasser, M. Lawrence; Moll, Victor H.; Vignat, Christophe
In 1998 Don Zagier introduced the modified Bernoulli numbers B∗nBn∗ and showed that they satisfy amusing variants of some properties of Bernoulli numbers. In particular, he studied the asymptotic behavior of B∗2nB2n∗, and also obtained an exact formula for them, the motivation for which came from the representation of B2nB2n in terms of the Riemann zeta function ζ(2n)ζ(2n). The modified Bernoulli numbers were recently generalized to Zagier polynomials B∗n(x)Bn∗(x). For 0<x<10<x<1, an exact formula for B∗2n(x)B2n∗(x) involving infinite series of Bessel function of the second kind and Chebyshev polynomials, that yields Zagier’s formula in a limiting case, is established here. Such series arise in diffraction theory. An analogous formula for B∗2n+1(x)B2n+1∗(x) is also presented. The 6-periodicity of B∗2n+1B2n+1∗ is deduced as a limiting case of it. These formulas are reminiscent of the Fourier expansions of Bernoulli polynomials. Some new results, for example, the one yielding the derivative of the Bessel function of the first kind with respect to its order as the Fourier coefficient of a function involving Chebyshev polynomials, are obtained in the course of proving these exact formulas. The asymptotic behavior of Zagier polynomials is also derived from them. Finally, a Zagier-type exact formula is obtained for B∗2n(−32)+B∗2nB2n∗(−32)+B2n∗.
Sat, 01 Jul 2017 00:00:00 GMThttp://repository.iitgn.ac.in/handle/123456789/29812017-07-01T00:00:00Z