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{"Title": "Bessel Rayleigh sums", "Definition": "The Bessel Rayleigh sum is $\\sigma_n(\\nu)=\\sum_{k=1}^{\\infty}j_{\\nu,k}^{-2n}$ CITE{Rayleigh}, where $j_{\\nu,k}$ is the $k$th HREF{Zeros_of_Bessel_J_functions}[strictly positive zero] of the HREF{Values_of_Bessel_functions_of_the_first_kind}[cylindrical Bessel function $J_\\nu$], counted in increasing order from $k=1$.", "Parameters": {"nu": {"type": "Q", "display": "$\\nu$", "title": "Bessel order", "constraints": "$\\nu\\geq0$"}, "n": {"type": "Z", "display": "$n$", "title": "inverse-power moment order", "constraints": "$n\\geq1$"}}, "Comments": {"normalization": "For the selected orders, the eigenvalues are $j_{\\nu,k}^2$ for a unit-radius disk or three-dimensional ball with Dirichlet boundary conditions. Each radial spectrum is counted once, without angular degeneracy factors. Only strictly positive roots are inverted; a zero at the origin is excluded. The individual roots are tabulated in HREF{Zeros_of_Bessel_J_functions}[the table of zeros of Bessel functions of the first kind].", "angular-orders": "Separation of variables gives $\\nu=\\ell$ on the disk and $\\nu=\\ell+1/2$ on the three-dimensional ball, where $\\ell$ is the nonnegative angular order CITE{Applications} CITE{Spherical}. The regular radial factor is $r^{-(d-2)/2}J_{\\ell+(d-2)/2}(\\sqrt{\\lambda}\\,r)$ in dimension $d=2,3$. The Dirichlet condition at $r=1$ selects $\\lambda=j_{\\nu,k}^2$.", "convergence": "The positive roots grow linearly with their index $k$, so the sum converges for every integer $n\\geq1$. It is the trace of the $n$th power of the inverse radial operator.", "rationality": "The normalized Bessel series has rational coefficients for rational $\\nu\\geq0$. Its canonical product CITE{Product} yields rational inverse-power sums, as also follows from CITE{formula-recurrence}."}, "Tags": ["special values", "special functions", "analysis", "physics"], "Keywords": ["Rayleigh sums", "Rayleigh function", "Bessel zeta", "Bessel spectral zeta"], "Links": {"Rayleigh": {"title": "DLMF 10.21(xiii): Rayleigh function", "url": "https://dlmf.nist.gov/10.21#xiii"}, "Applications": {"title": "DLMF 10.73(i\u2013ii): cylindrical and spherical separation of variables", "url": "https://dlmf.nist.gov/10.73"}, "Spherical": {"title": "DLMF 10.47.3: spherical and cylindrical Bessel functions", "url": "https://dlmf.nist.gov/10.47.E3"}, "Product": {"title": "DLMF 10.21.15: canonical Bessel product", "url": "https://dlmf.nist.gov/10.21.E15"}, "Series": {"title": "DLMF 10.2.2: Bessel power series", "url": "https://dlmf.nist.gov/10.2.E2"}, "Elementary": {"title": "DLMF 10.16.1: elementary half-order functions", "url": "https://dlmf.nist.gov/10.16.E1"}, "EvenZeta": {"title": "DLMF 25.6.2: even-zeta identity", "url": "https://dlmf.nist.gov/25.6.E2"}}, "Data properties": {"type": "Q", "rigour": "exact", "complete": "no", "complete-note": "it holds every $1\\leq n\\leq12$ for $\\nu=\\ell+(d-2)/2$ with $d=2,3$ and $\\ell=0,1,2,3$; these orders are the first four angular sectors of the unit disk and three-dimensional ball CITE{Applications} CITE{Spherical}, and $n$ indexes inverse-operator moments; both upper bounds are editorial prefixes of these spectral families", "rigour details": "All 96 values were obtained in exact rational arithmetic by coefficient comparison in the logarithmic derivative of the normalized Bessel series CITE{Series} and its canonical product CITE{Product}. No decimal approximation or truncated root sum enters the values. A separate implementation using Python fractions agrees with the convolution recurrence CITE{GuptaMuldoon} on every entry. The first three moment formulas agree at all eight orders. Every half-order specialization agrees with independently computed HREF{Bernoulli_numbers}[Bernoulli numbers] and with the even-zeta formula evaluated at 120 decimal digits. Positivity, strict decrease and strict log convexity hold throughout the stored range. A supplementary numerical sum over 32 positive roots per order, with an explicit upper bound for the infinite tail, agrees with all 96 values. That check uses numerically indexed roots and does not certify 100 digits; the exact rational computation supplies the stated rigour. The attached checks include known nonzero controls and a deliberately perturbed value, and were repeated on the values read back from the database."}, "Display properties": {"number-header": "$\\sigma_n(\\nu)$"}, "Numbers": {"0": {"1": "1/4", "2": "1/32", "3": "1/192", "4": "11/12288", "5": "19/122880", "6": "473/17694720", "7": "229/49545216", "8": "101369/126835752960", "9": "946523/6849130659840", "10": "65467219/2739652263936000", "11": "249045899/60272349806592000", "12": "9921896851/13886749395438796800"}, "1/2": {"1": "1/6", "2": "1/90", "3": "1/945", "4": "1/9450", "5": "1/93555", "6": "691/638512875", "7": "2/18243225", "8": "3617/325641566250", "9": "43867/38979295480125", "10": "174611/1531329465290625", "11": "155366/13447856940643125", "12": "236364091/201919571963756521875"}, "1": {"1": "1/8", "2": "1/192", "3": "1/3072", "4": "1/46080", "5": "13/8847360", "6": "11/110100480", "7": "647/95126814720", "8": "1133/2446118092800", "9": "43213/1369826131968000", "10": "777013/361634098839552000", "11": "2540291/17358436744298496000", "12": "31489807/3159235487462326272000"}, "3/2": {"1": "1/10", "2": "1/350", "3": "1/7875", "4": "37/6063750", "5": "59/197071875", "6": "2753/186232921875", "7": "1654/2261399765625", "8": "8386459/231589949997656250", "9": "28033727/15632321624841796875", "10": "14529522883/163592245803969404296875", "11": "327092834/74360111729077001953125", "12": "47497385017/218018127581059225341796875"}, "2": {"1": "1/12", "2": "1/576", "3": "1/17280", "4": "7/3317760", "5": "11/139345920", "6": "797/267544166400", "7": "181/1605264998400", "8": "823/192631799808000", "9": "2059/12713698787328000", "10": "839309/136697689361350656000", "11": "12410417/53312098850926755840000", "12": "1264807099/143302921711291119697920000"}, "5/2": {"1": "1/14", "2": "1/882", "3": "1/33957", "4": "47/55621566", "5": "73/2920132215", "6": "8573/11467359208305", "7": "20542/915095264822739", "8": "1686527/2498210072966077470", "9": "2643559/130127354094791858805", "10": "10652488897/17420799529440260097519375", "11": "32220274/1750415263723662497836875", "12": "60024688925533/108322737039493984681444624478625"}, "3": {"1": "1/16", "2": "1/1280", "3": "1/61440", "4": "13/34406400", "5": "1/110100480", "6": "263/1189085184000", "7": "103/19025362944000", "8": "15563/117196235735040000", "9": "55019/16876257945845760000", "10": "127753/1595573478516326400000", "11": "3866053/1965746525532114124800000", "12": "205133491/4246012495149366509568000000"}, "7/2": {"1": "1/18", "2": "1/1782", "3": "1/104247", "4": "19/103204530", "5": "29/7895146545", "6": "14363/193060018464885", "7": "18442/12162781163287755", "8": "8581361/276946527088062181350", "9": "256771367/405034295866290940224375", "10": "8846081929/681672719942967652397623125", "11": "4297537526/16174234536828596115979966875", "12": "733790620664989/134868417981012754341003822328081875"}}, "Formulas": {"formula-recurrence": "$\\sigma_1(\\nu)=\\frac{1}{4(\\nu+1)}$ and $\\displaystyle \\sigma_n(\\nu)=\\frac{1}{\\nu+n}\\sum_{r=1}^{n-1}\\sigma_r(\\nu)\\sigma_{n-r}(\\nu)$ for $n\\geq2$ CITE{GuptaMuldoon}, equations (3) and (4).", "formula-low-orders": "$\\displaystyle \\sigma_2(\\nu)=\\frac{1}{16(\\nu+1)^2(\\nu+2)},\\qquad \\sigma_3(\\nu)=\\frac{1}{32(\\nu+1)^3(\\nu+2)(\\nu+3)}$.", "formula-half-order": "$\\displaystyle \\sigma_n(1/2)=\\frac{\\zeta(2n)}{\\pi^{2n}}=\\frac{(-1)^{n+1}2^{2n-1}B_{2n}}{(2n)!}$ for $n\\geq1$, where $\\zeta$ is the HREF{Values_of_the_Riemann_zeta_function_at_rational_numbers}[Riemann zeta function] and $B_{2n}$ is a HREF{Bernoulli_numbers}[Bernoulli number]. The zeros are $j_{1/2,k}=k\\pi$ by CITE{Elementary}; the even-zeta evaluation is CITE{EvenZeta}."}, "References": {"GuptaMuldoon": {"bib": "D. P. Gupta and M. E. Muldoon, Riccati equations and convolution formulae for functions of Rayleigh type, Journal of Physics A: Mathematical and General 33 (2000), 1363\u20131368.", "arxiv": "math/9910128", "doi": "10.1088/0305-4470/33/7/306"}}, "Similar tables": {"Zeros_of_Bessel_J_functions": "The roots whose inverse even powers define these sums.", "Bernoulli_numbers": "The numbers $B_{2n}$ give the exact specialization at $\\nu=1/2$.", "Values_of_the_Riemann_zeta_function_at_rational_numbers": "The specialization at $\\nu=1/2$ is $\\zeta(2n)/\\pi^{2n}$."}, "Programs": {"program-python": {"language": "Python", "code": "from fractions import Fraction\n\ndef rayleigh(nu, n):\n nu = Fraction(nu)\n if nu < 0 or n < 1:\n raise ValueError(\"Require nu >= 0 and integer n >= 1\")\n s = [Fraction(0), Fraction(1, 4) / (nu + 1)]\n for m in range(2, n + 1):\n s.append(sum(s[k] * s[m-k] for k in range(1, m)) / (nu + m))\n return s[n]\n\nrayleigh(\"1/2\", 12)\n"}}}