Kil Hyun Kwon

Orthogonal polynomial solutions of linear ordinary differential equations

An electronic timepiece includes a reversible stepping motor to drive rotatable hands to provide a time display. In a first preferred embodiment, the electronic timepiece includes a circuit means to generate alternating current pulses having an increased pulse width or an increased amplitude to drive the stepping motor with an increased driving current during high speed time correction when a manually operable external control member is actuated during time correction. In a second preferred embodiment, the electronic timepiece comprises a clockwise correction switch, a counter-clockwise correction switch and circuit means for generating first and second alternating current pulses of first and second pulse widths at a predetermined frequency higher than normal driving pulses to drive a stepping motor in clockwise and counter-clockwise directions, respectively, to perform clockwise and counter-clockwise corrections when the clockwise and counter-clockwise correction switches are actuated during time corrections, respectively. The pulse width of the second alternating current pulses is selected to be smaller than that of the normal driving pulses.

Orthogonal polynomials in two variables and second-order partial differential equations

Abstract We study the second-order partial differential equations L [ u ] = Au xx +22 Bu xy + Cu yy + Du x + Eu y = λ n u , which have orthogonal polynomials in two variables as solutions. By using formal functional calculus on moment functionals, we first give new simpler proofs and improvements of the results by Krall and Sheffer and Littlejohn. We then give a two-variable version of Al-Salam and Chiharas characterization of classical orthogonal polynomials in one variable. We also study in detail the case when L [·] belongs to the basic class, that is, A y = C x = 0. In particular, we characterize all such differential equations which have a product of two classical orthogonal polynomials in one variable as solutions.

Characterizations of Classical Orthogonal Polynomials

We give a simple unified proof and an extension of some of the characterization theorems of classical orthogonal polynomials of Jacobi, Bessel, Laguerre, and Hermite. In particular, we prove that the only orthogonal polynomials whose derivatives form a weak orthogonal polynomial set are the classical orthogonal polynomials.

Orthogonal polynomial eigenfunctions of second-order partial differerential equations

In this paper, we show that for several second-order partial differential equations L[u] = A(x, y)uxx + 2B(x, y)uxy + C(x, y)uyy + D(x, y)ux + E(x, y)uy = AtU which have orthogonal polynomial eigenfunctions, these polynomials can be expressed as a product of two classical orthogonal polynomials in one variable. This is important since, otherwise, it is very difficult to explicitly find formulas for these polynomial solutions. From this observation and characterization, we are able to produce additional examples of such orthogonal polynomials together with their orthogonality that widens the class found by H. L. Krall and Sheffer in their seminal work in 1967. Moreover, from our approach, we can answer some open questions raised by Krall and Sheffer.

Partial differential equations having orthogonal polynomial solutions

Abstract We show that if a second order partial differential equation: L[u]:= Auxx + 2Buxy + Cuyy + Dux + Euy = λnu has orthogonal polynomial solutions, then the differential operator L[·] must be symmetrizable and can not be parabolic in any nonempty open subset of the plane. We also find Rodrigues type formula for orthogonal polynomial solutions of such differential equations.

Differential equations having orthogonal polynomial solutions

Abstract Necessary and sufficient conditions for an orthogonal polynomial system (OPS) to satisfy a differential equation with polynomial coefficients of the form (∗) L N [y] = ∑ i=1 N l i (x)y (i) (x) = λ n y(x) were found by H.L. Krall. Here, we find new necessary conditions for the equation (∗) to have an OPS of solutions as well as some other interesting applications. In particular, we obtain necessary and sufficient conditions for a distribution w(x) to be an orthogonalizing weight for such an OPS and investigate the structure of w(x). We also show that if the equation (∗) has an OPS of solutions, which is orthogonal relative to a distribution w(x), then the differential operator LN[·] in (∗) must be symmetrizable under certain conditions on w(x).

Real orthogonalizing weights for Bessel polynomials

Abstract We construct real orthogonalizing weights of bounded variation for the generalized Bessel polynomials.

Some Extension of the bessel-type orthogonal polynomials

We consider the perturbation of the classical Bessel moment functional by the addition of the linear functional . We give necessary and sufficient conditions in order for this functional to be a quasi-definite functional. In such a situation we analyze the corresponding sequence of monic orthogonal polynomials . In particular, a hypergeometric representation (4 F 2) for them is obtained. Furthermore, we deduce a relation between the corresponding Jacobi matrices, as well as the asymptotic behavior of the rario , outside of the closed contour Γ containing the origin and the difference between the new polynomials and the classical ones, inside Γ.

A characterization of Hermite polynomials

We show that for any orthogonal polynomials {P,(x)}~o satisfying a spectral type differential equation of order N (>~2) N LN[y](x) = ~ &(x)y(i)(x) = 2,y(x), i--I {Pn(x)}~0 must be essentially Hermite polynomials if and only if the leading coefficient EN(x) is a nonzero constant.

Differential equations of infinite order for Sobolev-type orthogonal polynomials

Abstract Assume that Pn(x)n=0∞ are orthogonal polynomials relative to a quasi-definite moment functional σ, which satisfy a differential equation of spectral type of order D (2⩽D⩽∞)where l i (x) are polynomials of degree ⩽i. Let be the symmetric bilinear form of discrete Sobolev type defined by (p,q) = 〈σ, pq〉 + Np(k)(c)q(k)(c) where N(≠0) and c are real constants, k is a non-negative integer, and p and q are polynomials. We first give a necessary and sufficient condition for to be quasi-definite and then show: If is quasi-definite, then the corresponding Sobolev-type orthogonal polynomials RnN,k;c(x)n=0∞ satisfy a differential equation of infinite order of the form φ(p,q)+Np (k) (c)q (k) (c) where ai(x)i=0∞ are polynomials of degree ⩽i, independent of n except a0(x) := a0(x,n). We also discuss conditions 3nder which such a differential equation is of finite order when σ is positive-definite, D