limiting form as a Bessel function
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1: 18.11 Relations to Other Functions
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Laguerre
…2: 18.34 Bessel Polynomials
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18.34.8
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3: 10.24 Functions of Imaginary Order
§10.24 Functions of Imaginary Order
… ►In consequence of (10.24.6), when is large and comprise a numerically satisfactory pair of solutions of (10.24.1); compare §2.7(iv). Also, in consequence of (10.24.7)–(10.24.9), when is small either and or and comprise a numerically satisfactory pair depending whether or . … ►4: 10.28 Wronskians and Cross-Products
5: 10.45 Functions of Imaginary Order
§10.45 Functions of Imaginary Order
… ►The corresponding result for is given by … ►In consequence of (10.45.5)–(10.45.7), and comprise a numerically satisfactory pair of solutions of (10.45.1) when is large, and either and , or and , comprise a numerically satisfactory pair when is small, depending whether or . … ►6: 10.5 Wronskians and Cross-Products
7: 10.2 Definitions
§10.2 Definitions
►§10.2(i) Bessel’s Equation
… ►Bessel Function of the First Kind
… ►Bessel Function of the Second Kind (Weber’s Function)
… ►Bessel Functions of the Third Kind (Hankel Functions)
…8: 33.10 Limiting Forms for Large or Large
§33.10 Limiting Forms for Large or Large
►§33.10(i) Large
… ►§33.10(ii) Large Positive
… ►§33.10(iii) Large Negative
…9: 37.12 Orthogonal Polynomials on Quadratic Surfaces
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►The bilinear form
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►Formula (37.12.5) is called an addition formula if there is a closed-form expression for the left-hand side.
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►The reproducing kernel in (37.12.5) satisfies for weight function (37.12.6) a closed-form formula, given as a four-fold integral involving (defined in (37.11.28)), see (Xu, 2020, Theorem 8.2).
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►where is the Bessel function §10.2(ii); moreover, when , the identity (37.12.5) holds under the limit relation (37.14.14).
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