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Bessel Moment Polynomials |
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PP: 1371-1375 |
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doi:10.18576/amis/200519
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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 study the family of even moments of squared Bessel functions of the first kind,
∞
Pn(x) = ∑ k2n Jk2(2x), n ≥ 1.
k=0
By means of the classical generating–function identity ∑∞k=−∞ Jk2(a)eiky = J0(2asiny/2), we prove that each Pn is an even polynomial
of degree 2n in x, or equivalently a polynomial pn(u) of degree n in the variable u = x2. We derive an explicit formula for the coefficients an,m and establish a three–term recurrence relation. The first few polynomials are
p1(u) = u,
p2(u) = 3u2 +u,
p3(u) = 10u3 +15u2 +u,
p4(u) = 35u4 +140u3 +63u2 +u,
p5 (u) = 126u5 + 1050u4 + 1470u3 + 255u2 + u.
All formulas are verified numerically with tight truncation–error bounds. |
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