| Summary: | The Bessel function of the first kind <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>N</mi></msub><mfenced separators="" open="(" close=")"><mi>k</mi><mi>x</mi></mfenced></mrow></semantics></math></inline-formula> is expanded in a Fourier–Legendre series, as is the modified Bessel function of the first kind <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>I</mi><mi>N</mi></msub><mfenced separators="" open="(" close=")"><mi>k</mi><mi>x</mi></mfenced></mrow></semantics></math></inline-formula>. The purpose of these expansions in Legendre polynomials was not an attempt to rival established <i>numerical</i> methods for calculating Bessel functions but to provide a form for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>N</mi></msub><mfenced separators="" open="(" close=")"><mi>k</mi><mi>x</mi></mfenced></mrow></semantics></math></inline-formula> useful for <i>analytical</i> work in the area of strong laser fields, where analytical integration over scattering angles is essential. Despite their primary purpose, one can easily truncate the series at 21 terms to provide 33-digit accuracy that matches the IEEE extended precision in some compilers. The analytical theme is furthered by showing that infinite series of like-powered contributors (involving <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mo> </mo><mspace width="-0.166667em"></mspace></mrow><mn>1</mn></msub><msub><mi>F</mi><mn>2</mn></msub></mrow></semantics></math></inline-formula> hypergeometric functions) extracted from the Fourier–Legendre series may be summed, having values that are inverse powers of the eight primes <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>/</mo><mfenced separators="" open="(" close=")"><msup><mn>2</mn><mi>i</mi></msup><msup><mn>3</mn><mi>j</mi></msup><msup><mn>5</mn><mi>k</mi></msup><msup><mn>7</mn><mi>l</mi></msup><msup><mn>11</mn><mi>m</mi></msup><msup><mn>13</mn><mi>n</mi></msup><msup><mn>17</mn><mi>o</mi></msup><msup><mn>19</mn><mi>p</mi></msup></mfenced></mrow></semantics></math></inline-formula> multiplying powers of the coefficient <i>k</i>.
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