## Continuous ranking of zeros of special functions

Venue: | J. Math. Anal. Appl |

Citations: | 3 - 2 self |

### BibTeX

@ARTICLE{Muldoon_continuousranking,

author = {Martin E. Muldoon},

title = {Continuous ranking of zeros of special functions},

journal = {J. Math. Anal. Appl},

year = {},

volume = {343},

pages = {436--445}

}

### OpenURL

### Abstract

Abstract. We reexamine and continue the work of J. Vosmansky´y [23] on the concept of continuous ranking of zeros of certain special functions from the point of view of the transformation theory of second order linear differential equations. This leads to results on higher monotonicity of such zeros with respect to the rank and to the evaluation of some definite integrals. The applications are to Airy, Bessel and Hermite functions. 1.

### Citations

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14 |
Some recent results on the zeros of Bessel functions and orthogonal polynomials
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(Show Context)
Citation Context ...r, α and k are not really independent; they may be subsumed in a single variable κ = k − α/π. This was done by Elbert and Laforgia [8] in the case of Bessel functions. Their work is also described in =-=[6]-=- and [15]. To see that their approach is equivalent to Definition 1.1, we consider that the zeros xk(α) of y(x, α), 0 < α < π are the roots of the equation (2.11) y2(x)/y1(x) = cot α.sZEROS OF SPECIAL... |

13 | Higher monotonicity properties of certain Sturm–Liouville functions - Lorch, Szegő - 1963 |

11 |
Concavity of the zeros of Bessel functions
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(Show Context)
Citation Context ...been considered by many authors; see [2], [17]. The concept of continuous ranking of zeros of Bessel functions occurs in the work of Á. Elbert and A. Laforgia on zeros of Bessel functions starting in =-=[6]-=-; see [5, p. 67] for a summary and a recognition that the concept appeared already in the work of F. W. J. Olver [18]. J. Vosmansk´y [20] made the connection between the transformation theory and cont... |

8 |
Nicholson-type integrals for products of Gegenbauer functions and related topics, Theory and
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(Show Context)
Citation Context ...1(− 1 1 2λ, 2 , x2 ); v(x, λ) = 1 (6.5) φ(λ) = φ(λ) e−x2 /2x 1F1(− 1 2 � 1 2 Γ( 1 2 Γ( 1 2 (λ + 1)) (λ + 2)) 1 3 λ + 2 , 2 , x2 ); �1/2 . This choice ensures that W (y1, y2) = 1. Hλ(x), considered in =-=[4]-=-, [12] and [8], is a generalization to real λ of the Hermite polynomial to which it reduces in case λ = 0, 1, 2, . . . . The solution y1(x) is the principal solution of (6.1) at +∞; this follows from ... |

4 |
Inequalities and monotonicity properties for zeros of Hermite functions
- Elbert, Muldoon
- 1999
(Show Context)
Citation Context ..., x2 ); v(x, λ) = 1 (6.5) φ(λ) = φ(λ) e−x2 /2x 1F1(− 1 2 � 1 2 Γ( 1 2 Γ( 1 2 (λ + 1)) (λ + 2)) 1 3 λ + 2 , 2 , x2 ); �1/2 . This choice ensures that W (y1, y2) = 1. Hλ(x), considered in [4], [12] and =-=[8]-=-, is a generalization to real λ of the Hermite polynomial to which it reduces in case λ = 0, 1, 2, . . . . The solution y1(x) is the principal solution of (6.1) at +∞; this follows from the asymptotic... |

4 |
An upper bound for the largest zero of Hermite’s function with applications to subharmonic functions
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(Show Context)
Citation Context ... 1 2λ, 2 , x2 ); v(x, λ) = 1 (6.5) φ(λ) = φ(λ) e−x2 /2x 1F1(− 1 2 � 1 2 Γ( 1 2 Γ( 1 2 (λ + 1)) (λ + 2)) 1 3 λ + 2 , 2 , x2 ); �1/2 . This choice ensures that W (y1, y2) = 1. Hλ(x), considered in [4], =-=[12]-=- and [8], is a generalization to real λ of the Hermite polynomial to which it reduces in case λ = 0, 1, 2, . . . . The solution y1(x) is the principal solution of (6.1) at +∞; this follows from the as... |

3 |
Monotonicity properties of the zeros of Bessel functions
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- 1986
(Show Context)
Citation Context ...sents a fixed value of ν. (These and the other figures were produced with the aid of Maple.) Note that we get a straight line for ν = 1 2 . The figure also confirms the results of Elbert and Laforgia =-=[7]-=- that jνκ is a convex function of κ for 0 < ν < 1 2 concave function of κ for ν > 1 2 . 0 −∞ . and a In Figure 1, we do not go beyond κ = 1. For larger values of κ, the curves become increasingly indi... |

3 | Higher monotonicity properties and inequalities for zeros of Bessel functions - Gori, Laforgia, et al. - 1991 |

3 |
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(Show Context)
Citation Context ... k are not really independent; they may be subsumed in a single variable κ = k − α/π. This was done by Elbert and Laforgia [8] in the case of Bessel functions. Their work is also described in [6] and =-=[15]-=-. To see that their approach is equivalent to Definition 1.1, we consider that the zeros xk(α) of y(x, α), 0 < α < π are the roots of the equation (2.11) y2(x)/y1(x) = cot α.sZEROS OF SPECIAL FUNCTION... |

2 |
uvka, Linear Differential Transformations of the Second Order, translated by F
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(Show Context)
Citation Context ...ry, Bessel and Hermite functions. 1. Introduction The transformation theory of second order linear differential equations including the concept of first phase has been considered by many authors; see =-=[2]-=-, [17]. The concept of continuous ranking of zeros of Bessel functions occurs in the work of Á. Elbert and A. Laforgia on zeros of Bessel functions starting in [6]; see [5, p. 67] for a summary and a ... |

2 |
A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order
- Olver
- 1951
(Show Context)
Citation Context ...s in the work of Á. Elbert and A. Laforgia on zeros of Bessel functions starting in [6]; see [5, p. 67] for a summary and a recognition that the concept appeared already in the work of F. W. J. Olver =-=[18]-=-. J. Vosmansk´y [20] made the connection between the transformation theory and continuous ranking concepts. Our purpose here is to reexamine the connection between these ideas. We are thus able to obt... |

2 | Bounds on differences of adjacent zeros of Bessel functions and iterative relations between consecutive zeros - Segura |

2 |
Zeros of solutions of linear differential equations as continuous functions of the parameter k
- Vosmansk´y
- 1991
(Show Context)
Citation Context ...Elbert and A. Laforgia on zeros of Bessel functions starting in [6]; see [5, p. 67] for a summary and a recognition that the concept appeared already in the work of F. W. J. Olver [18]. J. Vosmansk´y =-=[20]-=- made the connection between the transformation theory and continuous ranking concepts. Our purpose here is to reexamine the connection between these ideas. We are thus able to obtain some new results... |

2 |
A new method for the evaluation of zeros of Besel functions and of other solutions of second-order differential equations
- Olver
- 1950
(Show Context)
Citation Context ...s in the work of Á. Elbert and A. Laforgia on zeros of Bessel functions starting in [8]; see [6, p. 67] for a summary and a recognition that the concept appeared already in the work of F. W. J. Olver =-=[20, 21]-=-. J. Vosmansk´y [23] made the connection between the transformation theory and continuous ranking concepts and gave applications to Bessel functions. Our purpose here is to reexamine the connection be... |

1 |
Hidden dimensions
- Butcher
(Show Context)
Citation Context ...y [19, Figure 1]. This gives the picture a three-dimensional appearance as does the alternation in thickness of the graph segments on the front and back of the cylinder; cf. the opening paragraphs in =-=[3]-=-. 5. Derivatives with respect to κ Differentiating (2.8) with respect to κ gives (5.1) x ′ (κ) = πp[x(κ)]. When applied to the equation (4.1) this gives a result mentioned by Elbert [5, (1.4)] (5.2) d... |

1 |
Continuous ranking of zeros of cylinder functions
- Lorch
(Show Context)
Citation Context ...wer restrictions on the interval (a, b) (it can be finite or infinite) and there is no need to restrict y1, y2 explicitly to be what Vosmansk´y calls principal pairs. Our title is inspired by that of =-=[16]-=-. 3. Airy functions The function Ai(−x) [1, §10.4] is a principal solution at −∞ of the Airy equation (3.1) y ′′ + xy = 0.s4 MARTIN E. MULDOON y 3 2.5 2 1.5 1 0.5 0 0.2 0.4 0.6 0.8 1 x Figure 1. Appro... |

1 |
Extension of a result of L
- Muldoon
- 1968
(Show Context)
Citation Context ...)[x ′ (κ)] 2 +p ′ (x)x ′′ (κ), . . . . By induction, one can show that, for each n, (−1) n x (n+1) (κ) is a sum of nonnegative terms. In the case of the stronger hypotheses (5.5), one can show, as in =-=[16]-=- that one of these terms is positive. Our theorem generalizes the case of Bessel functions (ν > 1/2) considered in [10, Corollary 3.3]. It extends to derivatives with respect to κ the special case λ =... |