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Closed Forms for Odd Moments of Squared Bessel Functions |
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PP: 19-30 |
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doi:10.18576/amis/210102
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Author(s) |
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He ́ctor M. Moya-Cessa,
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Abstract |
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| We investigate the odd moments of squared Bessel functions, S2n+1(x) = ∑∞k=0 k2n+1Jk2(2x), and derive a systematic finite representation valid for arbitrary nonnegative integer (n). Using Bessel-function recurrence relations and a telescoping construction, we show that each odd moment can be expressed in terms of the three basic products J02(2x), J12(2x), J0(2x)J1(2x), together
with a terminal contribution involving J0(2x)J2n(2x)+J1(2x)J2n+1(2x). The corresponding coefficient functions are polynomials in x2 and are generated by a finite algebraic procedure based on a polynomial antidifference operator. We derive the associated difference equation, establish the polynomial character and uniqueness of its solutions, and obtain explicit coefficient formulas for the first several odd moments. The resulting construction reproduces the known expressions for the first and third moments and provides, in a unified way, closed finite representations for higher odd moments. This approach also yields a convenient algebraic framework for generating moments of arbitrarily high order without evaluating the original infinite series directly. |
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