On a Generalisation of Legendre’s Associated Differential Equation. I

Kuipers, L.; Meulenbeld, B.

doi:10.1016/s1385-7258(57)50057-7

Cited by 12 publications

(6 citation statements)

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“…Classical Legendre functions [1,3] are recovered from the formulae (1), (2), (4) and (5) for r = 0. Let us give some basic properties of these generalized functions.…”

Section: Generalized Associated Legendre Functionsmentioning

confidence: 99%

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Some results involving generalized associated Legendre functions

Kalla

Virchenko

Lisetska

2012

Integral Transforms and Special Functions

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In this paper, a new generalization of the associated Legendre functions of the first and the second kinds is introduced using the r-generalized Gauss hypergeometric function. The basic properties of these functions, in particular, some recurrence relations and differential and integral representations, are given. The Whipple formulae are established. A new generalization of the classical Mehler-Fock integral transform is constructed, and the inversion formula is proved. Some new integrals involving the functions τ,β r P μ ν (t) and τ,β r P m,n k (z) are evaluated.

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“…Classical Legendre functions [1,3] are recovered from the formulae (1), (2), (4) and (5) for r = 0. Let us give some basic properties of these generalized functions.…”

Section: Generalized Associated Legendre Functionsmentioning

confidence: 99%

“…Now, there exist various generalizations of the Legendre and the associated Legendre functions. In 1957, Kuipers and Meulenbeld [3] introduced a class of generalized associated Legendre functions of the first and the second kinds P m,n k (z) and Q m,n k (z), respectively. These functions satisfy the differential equation:…”

Section: Introductionmentioning

confidence: 99%

Some results involving generalized associated Legendre functions

Kalla

Virchenko

Lisetska

2012

Integral Transforms and Special Functions

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“…were introduced as linearly independent solutions of the following generalized Legendre differential equation [11]:…”

Section: The Generalized Associated Legendre Functionsmentioning

confidence: 99%

“…In 1957 Kuipers and Meulenbeld [11] introduced into consideration the generalized associated Legendre functions of the first and the second kind P m,n k (z) and Q m,n k (z) respectively, see [11], [16], [19].…”

Section: Introductionmentioning

confidence: 99%

The generalized associated Legendre functions

Virchenko¹,

Fedotova²

2001

Generalized Associated Legendre Functions and Their Applications

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“…We shall consider some integral operators with the generalized associated Legendre's functions P m,n k (z). The generalized associated functions P m,n k (z) and Q m,n k (z) are two linear-independent solutions of the following differential equation [10]:…”

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confidence: 99%