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Bessel polynomials
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<h2 id="properties">Properties</h2> <h3>Definition in terms of Bessel functions</h3> <p>The Bessel polynomial may also be defined using <a href="/facts/Bessel_function/hSyvFPqC">Bessel functions</a> from which the polynomial draws its name. </p> y n ( x ) = x n θ n ( 1 / x ) {\displaystyle y_{n}(x)=\,x^{n}\theta _{n}(1/x)\,} y n ( x ) = 2 π x e 1 / x K n + 1 2 ( 1 / x ) {\displaystyle y_{n}(x)={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{n+{\frac {1}{2}}}(1/x)} θ n ( x ) = 2 π x n + 1 / 2 e x K n + 1 2 ( x ) {\displaystyle \theta _{n}(x)={\sqrt {\frac {2}{\pi }}}\,x^{n+1/2}e^{x}K_{n+{\frac {1}{2}}}(x)} <p>where <i>K</i><i>n</i>(<i>x</i>) is a <a href="/facts/Bessel_function/hSyvFPqC">modified Bessel function of the second kind</a>, <i>y</i><i>n</i>(<i>x</i>) is the ordinary polynomial, and <i>θ</i><i>n</i>(<i>x</i>) is the reverse polynomial .<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a>: 7, 34 For example:<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p> y 3 ( x ) = 15 x 3 + 15 x 2 + 6 x + 1 = 2 π x e 1 / x K 3 + 1 2 ( 1 / x ) {\displaystyle y_{3}(x)=15x^{3}+15x^{2}+6x+1={\sqrt {\frac {2}{\pi x}}}\,e^{1/x}K_{3+{\frac {1}{2}}}(1/x)} <h3>Definition as a hypergeometric function</h3> <p>The Bessel polynomial may also be defined as a <a href="/facts/Confluent_hypergeometric_function/D0FxgWUa">confluent hypergeometric function</a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>: 8 </p> y n ( x ) = 2 F 0 ( − n , n + 1 ; ; − x / 2 ) = ( 2 x ) − n U ( − n , − 2 n , 2 x ) = ( 2 x ) n + 1 U ( n + 1 , 2 n + 2 , 2 x ) . {\displaystyle y_{n}(x)=\,_{2}F_{0}(-n,n+1;;-x/2)=\left({\frac {2}{x}}\right)^{-n}U\left(-n,-2n,{\frac {2}{x}}\right)=\left({\frac {2}{x}}\right)^{n+1}U\left(n+1,2n+2,{\frac {2}{x}}\right).} <p>A similar expression holds true for the generalized Bessel polynomials (see below):<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a>: 35 </p> y n ( x ; a , b ) = 2 F 0 ( − n , n + a − 1 ; ; − x / b ) = ( b x ) n + a − 1 U ( n + a − 1 , 2 n + a , b x ) . {\displaystyle y_{n}(x;a,b)=\,_{2}F_{0}(-n,n+a-1;;-x/b)=\left({\frac {b}{x}}\right)^{n+a-1}U\left(n+a-1,2n+a,{\frac {b}{x}}\right).} <p>The reverse Bessel polynomial may be defined as a generalized <a href="/facts/Laguerre_polynomial/7RM3ypWl">Laguerre polynomial</a>: </p> θ n ( x ) = n ! ( − 2 ) n L n − 2 n − 1 ( 2 x ) {\displaystyle \theta _{n}(x)={\frac {n!}{(-2)^{n}}}\,L_{n}^{-2n-1}(2x)} <p>from which it follows that it may also be defined as a hypergeometric function: </p> θ n ( x ) = ( − 2 n ) n ( − 2 ) n 1 F 1 ( − n ; − 2 n ; 2 x ) {\displaystyle \theta _{n}(x)={\frac {(-2n)_{n}}{(-2)^{n}}}\,\,_{1}F_{1}(-n;-2n;2x)} <p>where (−2<i>n</i>)<i>n</i> is the <a href="/facts/Pochhammer_symbol/sAxnmGoC">Pochhammer symbol</a> (rising factorial). </p> <h3>Generating function</h3> <p>The Bessel polynomials, with index shifted, have the generating function </p> ∑ n = 0 ∞ 2 π x n + 1 2 e x K n − 1 2 ( x ) t n n ! = 1 + x ∑ n = 1 ∞ θ n − 1 ( x ) t n n ! = e x ( 1 − 1 − 2 t ) . {\displaystyle \sum _{n=0}^{\infty }{\sqrt {\frac {2}{\pi }}}x^{n+{\frac {1}{2}}}e^{x}K_{n-{\frac {1}{2}}}(x){\frac {t^{n}}{n!}}=1+x\sum _{n=1}^{\infty }\theta _{n-1}(x){\frac {t^{n}}{n!}}=e^{x(1-{\sqrt {1-2t}})}.} <p>Differentiating with respect to t {\displaystyle t} , cancelling x {\displaystyle x} , yields the generating function for the polynomials { θ n } n ≥ 0 {\displaystyle \{\theta _{n}\}_{n\geq 0}} </p> ∑ n = 0 ∞ θ n ( x ) t n n ! = 1 1 − 2 t e x ( 1 − 1 − 2 t ) . {\displaystyle \sum _{n=0}^{\infty }\theta _{n}(x){\frac {t^{n}}{n!}}={\frac {1}{\sqrt {1-2t}}}e^{x(1-{\sqrt {1-2t}})}.} <p>Similar generating function exists for the y n {\displaystyle y_{n}} polynomials as well:<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>: 106 </p> ∑ n = 0 ∞ y n − 1 ( x ) t n n ! = exp ( 1 − 1 − 2 x t x ) . {\displaystyle \sum _{n=0}^{\infty }y_{n-1}(x){\frac {t^{n}}{n!}}=\exp \left({\frac {1-{\sqrt {1-2xt}}}{x}}\right).} <p>Upon setting t = z − x z 2 / 2 {\displaystyle t=z-xz^{2}/2} , one has the following representation for the <a href="/facts/Exponential_function/ewlYxvR2">exponential function</a>:<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>: 107 </p> e z = ∑ n = 0 ∞ y n − 1 ( x ) ( z − x z 2 / 2 ) n n ! . {\displaystyle e^{z}=\sum _{n=0}^{\infty }y_{n-1}(x){\frac {(z-xz^{2}/2)^{n}}{n!}}.} <h3>Recursion</h3> <p>The Bessel polynomial may also be defined by a recursion formula: </p> y 0 ( x ) = 1 {\displaystyle y_{0}(x)=1\,} y 1 ( x ) = x + 1 {\displaystyle y_{1}(x)=x+1\,} y n ( x ) = ( 2 n − 1 ) x y n − 1 ( x ) + y n − 2 ( x ) {\displaystyle y_{n}(x)=(2n\!-\!1)x\,y_{n-1}(x)+y_{n-2}(x)\,} <p>and </p> θ 0 ( x ) = 1 {\displaystyle \theta _{0}(x)=1\,} θ 1 ( x ) = x + 1 {\displaystyle \theta _{1}(x)=x+1\,} θ n ( x ) = ( 2 n − 1 ) θ n − 1 ( x ) + x 2 θ n − 2 ( x ) {\displaystyle \theta _{n}(x)=(2n\!-\!1)\theta _{n-1}(x)+x^{2}\theta _{n-2}(x)\,} <h3>Differential equation</h3> <p>The Bessel polynomial obeys the following differential equation: </p> x 2 d 2 y n ( x ) d x 2 + 2 ( x + 1 ) d y n ( x ) d x − n ( n + 1 ) y n ( x ) = 0 {\displaystyle x^{2}{\frac {d^{2}y_{n}(x)}{dx^{2}}}+2(x\!+\!1){\frac {dy_{n}(x)}{dx}}-n(n+1)y_{n}(x)=0} <p>and </p> x d 2 θ n ( x ) d x 2 − 2 ( x + n ) d θ n ( x ) d x + 2 n θ n ( x ) = 0 {\displaystyle x{\frac {d^{2}\theta _{n}(x)}{dx^{2}}}-2(x\!+\!n){\frac {d\theta _{n}(x)}{dx}}+2n\,\theta _{n}(x)=0} <h3>Orthogonality</h3> <p>The Bessel polynomials are orthogonal with respect to the weight e − 2 / x {\displaystyle e^{-2/x}} integrated over the unit circle of the complex plane.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>: 104 In other words, if n ≠ m {\displaystyle n\neq m} , </p><p> ∫ 0 2 π y n ( e i θ ) y m ( e i θ ) i e i θ d θ = 0 {\displaystyle \int _{0}^{2\pi }y_{n}\left(e^{i\theta }\right)y_{m}\left(e^{i\theta }\right)ie^{i\theta }\mathrm {d} \theta =0} </p> <h2 id="generalization">Generalization</h2> <h3>Explicit form</h3> <p>A generalization of the Bessel polynomials have been suggested in literature, as following: </p> y n ( x ; α , β ) := ( − 1 ) n n ! ( x β ) n L n ( − 1 − 2 n − α ) ( β x ) , {\displaystyle y_{n}(x;\alpha ,\beta ):=(-1)^{n}n!\left({\frac {x}{\beta }}\right)^{n}L_{n}^{(-1-2n-\alpha )}\left({\frac {\beta }{x}}\right),} <p>the corresponding reverse polynomials are </p> θ n ( x ; α , β ) := n ! ( − β ) n L n ( − 1 − 2 n − α ) ( β x ) = x n y n ( 1 x ; α , β ) . {\displaystyle \theta _{n}(x;\alpha ,\beta ):={\frac {n!}{(-\beta )^{n}}}L_{n}^{(-1-2n-\alpha )}(\beta x)=x^{n}y_{n}\left({\frac {1}{x}};\alpha ,\beta \right).} <p>The explicit coefficients of the y n ( x ; α , β ) {\displaystyle y_{n}(x;\alpha ,\beta )} polynomials are:<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>: 108 </p> y n ( x ; α , β ) = ∑ k = 0 n ( n k ) ( n + k + α − 2 ) k _ ( x β ) k . {\displaystyle y_{n}(x;\alpha ,\beta )=\sum _{k=0}^{n}{\binom {n}{k}}(n+k+\alpha -2)^{\underline {k}}\left({\frac {x}{\beta }}\right)^{k}.} <p>Consequently, the θ n ( x ; α , β ) {\displaystyle \theta _{n}(x;\alpha ,\beta )} polynomials can explicitly be written as follows: </p> θ n ( x ; α , β ) = ∑ k = 0 n ( n k ) ( 2 n − k + α − 2 ) n − k _ x k β n − k . {\displaystyle \theta _{n}(x;\alpha ,\beta )=\sum _{k=0}^{n}{\binom {n}{k}}(2n-k+\alpha -2)^{\underline {n-k}}{\frac {x^{k}}{\beta ^{n-k}}}.} <p>For the weighting function </p> ρ ( x ; α , β ) := 1 F 1 ( 1 , α − 1 , − β x ) {\displaystyle \rho (x;\alpha ,\beta ):={}_{1}F_{1}\left(1,\alpha -1,-{\frac {\beta }{x}}\right)} <p>they are orthogonal, for the relation </p> 0 = ∮ c ρ ( x ; α , β ) y n ( x ; α , β ) y m ( x ; α , β ) d x {\displaystyle 0=\oint _{c}\rho (x;\alpha ,\beta )y_{n}(x;\alpha ,\beta )y_{m}(x;\alpha ,\beta )\,\mathrm {d} x} <p>holds for <i>m</i> ≠ <i>n</i> and <i>c</i> a curve surrounding the 0 point. </p><p>They specialize to the Bessel polynomials for α = β = 2, in which situation ρ(<i>x</i>) = exp(−2/<i>x</i>). </p> <h3>Inverse Formulae</h3> <p>Powers of x {\displaystyle x} are expressed in terms of the generalized Bessel polynomials from the inverse connection formulae which have applications in <a href="/facts/Change_of_basis/Hnu1Spcc">change of basis</a> to these polynomials.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> x n = ∑ k = 0 n α ( n , k , α , β ) y n − k ( x ; α , β ) {\displaystyle x^{n}=\sum _{k=0}^{n}\alpha (n,k,\alpha ,\beta )\ y_{n-k}(x;\alpha ,\beta )} <p>where α ( n , k , α , β ) = ( n k ) ( − 1 ) k β n ( 2 ( n − k ) + α − 1 ) ( n − k + α − 1 ) n + 1 {\displaystyle \alpha (n,k,\alpha ,\beta )={n \choose k}{\frac {(-1)^{k}\beta ^{n}(2(n-k)+\alpha -1)}{(n-k+\alpha -1)_{n+1}}}} and n ≥ 0 {\displaystyle n\geq 0} . </p><p>Similarly, for the reverse generalized Bessel polynomials </p> x n = ∑ k = 0 n α 1 ( n , k , α , β ) θ n − k ( x ; α , β ) {\displaystyle x^{n}=\sum _{k=0}^{n}\alpha _{1}(n,k,\alpha ,\beta )\ \theta _{n-k}(x;\alpha ,\beta )} <p>where α 1 ( n , k , α , β ) = ( − 1 ) k ( n k ) β k ( n + α − 1 ) ( n + α − 2 k ) k − 1 {\displaystyle \alpha _{1}(n,k,\alpha ,\beta )=(-1)^{k}{\frac {n \choose k}{\beta ^{k}}}(n+\alpha -1)(n+\alpha -2k)_{k-1}} and n ≥ 0 {\displaystyle n\geq 0} . </p> <h3>Rodrigues formula for Bessel polynomials</h3> <p>The Rodrigues formula for the Bessel polynomials as particular solutions of the above differential equation is : </p> B n ( α , β ) ( x ) = a n ( α , β ) x α e − β x ( d d x ) n ( x α + 2 n e − β x ) {\displaystyle B_{n}^{(\alpha ,\beta )}(x)={\frac {a_{n}^{(\alpha ,\beta )}}{x^{\alpha }e^{-{\frac {\beta }{x}}}}}\left({\frac {d}{dx}}\right)^{n}(x^{\alpha +2n}e^{-{\frac {\beta }{x}}})} <p>where <i>a</i>(α, β)<i>n</i> are normalization coefficients. </p> <h3>Associated Bessel polynomials</h3> <p>According to this generalization we have the following generalized differential equation for associated Bessel polynomials: </p> x 2 d 2 B n , m ( α , β ) ( x ) d x 2 + [ ( α + 2 ) x + β ] d B n , m ( α , β ) ( x ) d x − [ n ( α + n + 1 ) + m β x ] B n , m ( α , β ) ( x ) = 0 {\displaystyle x^{2}{\frac {d^{2}B_{n,m}^{(\alpha ,\beta )}(x)}{dx^{2}}}+[(\alpha +2)x+\beta ]{\frac {dB_{n,m}^{(\alpha ,\beta )}(x)}{dx}}-\left[n(\alpha +n+1)+{\frac {m\beta }{x}}\right]B_{n,m}^{(\alpha ,\beta )}(x)=0} <p>where 0 ≤ m ≤ n {\displaystyle 0\leq m\leq n} . The solutions are, </p> B n , m ( α , β ) ( x ) = a n , m ( α , β ) x α + m e − β x ( d d x ) n − m ( x α + 2 n e − β x ) {\displaystyle B_{n,m}^{(\alpha ,\beta )}(x)={\frac {a_{n,m}^{(\alpha ,\beta )}}{x^{\alpha +m}e^{-{\frac {\beta }{x}}}}}\left({\frac {d}{dx}}\right)^{n-m}(x^{\alpha +2n}e^{-{\frac {\beta }{x}}})} <h2 id="zeros">Zeros</h2> <p>If one denotes the zeros of y n ( x ; α , β ) {\displaystyle y_{n}(x;\alpha ,\beta )} as α k ( n ) ( α , β ) {\displaystyle \alpha _{k}^{(n)}(\alpha ,\beta )} , and that of the θ n ( x ; α , β ) {\displaystyle \theta _{n}(x;\alpha ,\beta )} by β k ( n ) ( α , β ) {\displaystyle \beta _{k}^{(n)}(\alpha ,\beta )} , then the following estimates exist:<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>: 82 </p> 2 n ( n + α − 1 ) ≤ α k ( n ) ( α , 2 ) ≤ 2 n + α − 1 , {\displaystyle {\frac {2}{n(n+\alpha -1)}}\leq \alpha _{k}^{(n)}(\alpha ,2)\leq {\frac {2}{n+\alpha -1}},} <p>and </p> n + α − 1 2 ≤ β k ( n ) ( α , 2 ) ≤ n ( n + α − 1 ) 2 , {\displaystyle {\frac {n+\alpha -1}{2}}\leq \beta _{k}^{(n)}(\alpha ,2)\leq {\frac {n(n+\alpha -1)}{2}},} <p>for all α ≥ 2 {\displaystyle \alpha \geq 2} . Moreover, all these zeros have negative real part. </p><p>Sharper results can be said if one resorts to more powerful theorems regarding the estimates of zeros of polynomials (more concretely, the Parabola Theorem of Saff and Varga, or differential equations techniques).<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a>: 88 <a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> One result is the following:<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p> 2 2 n + α − 2 3 ≤ α k ( n ) ( α , 2 ) ≤ 2 n + α − 1 . {\displaystyle {\frac {2}{2n+\alpha -{\frac {2}{3}}}}\leq \alpha _{k}^{(n)}(\alpha ,2)\leq {\frac {2}{n+\alpha -1}}.} <h2 id="particular-values">Particular values</h2> <p>The Bessel polynomials y n ( x ) {\displaystyle y_{n}(x)} up to n = 5 {\displaystyle n=5} are<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> </p> y 0 ( x ) = 1 y 1 ( x ) = x + 1 y 2 ( x ) = 3 x 2 + 3 x + 1 y 3 ( x ) = 15 x 3 + 15 x 2 + 6 x + 1 y 4 ( x ) = 105 x 4 + 105 x 3 + 45 x 2 + 10 x + 1 y 5 ( x ) = 945 x 5 + 945 x 4 + 420 x 3 + 105 x 2 + 15 x + 1 {\displaystyle {\begin{aligned}y_{0}(x)&=1\\y_{1}(x)&=x+1\\y_{2}(x)&=3x^{2}+3x+1\\y_{3}(x)&=15x^{3}+15x^{2}+6x+1\\y_{4}(x)&=105x^{4}+105x^{3}+45x^{2}+10x+1\\y_{5}(x)&=945x^{5}+945x^{4}+420x^{3}+105x^{2}+15x+1\end{aligned}}} <p>No Bessel polynomial can be factored into lower degree polynomials with rational coefficients.<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> The reverse Bessel polynomials are obtained by reversing the coefficients. Equivalently, θ k ( x ) = x k y k ( 1 / x ) {\textstyle \theta _{k}(x)=x^{k}y_{k}(1/x)} . This results in the following: </p> θ 0 ( x ) = 1 θ 1 ( x ) = x + 1 θ 2 ( x ) = x 2 + 3 x + 3 θ 3 ( x ) = x 3 + 6 x 2 + 15 x + 15 θ 4 ( x ) = x 4 + 10 x 3 + 45 x 2 + 105 x + 105 θ 5 ( x ) = x 5 + 15 x 4 + 105 x 3 + 420 x 2 + 945 x + 945 {\displaystyle {\begin{aligned}\theta _{0}(x)&=1\\\theta _{1}(x)&=x+1\\\theta _{2}(x)&=x^{2}+3x+3\\\theta _{3}(x)&=x^{3}+6x^{2}+15x+15\\\theta _{4}(x)&=x^{4}+10x^{3}+45x^{2}+105x+105\\\theta _{5}(x)&=x^{5}+15x^{4}+105x^{3}+420x^{2}+945x+945\\\end{aligned}}} <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Bessel_function/hSyvFPqC">Bessel function</a></li> <li><a href="/facts/Neumann_polynomial/XXDhjXn5">Neumann polynomial</a></li> <li><a href="/facts/Lommel_polynomial/F1WsmQBu">Lommel polynomial</a></li> <li><a href="/facts/Hankel_transform/2UpX2jL8">Hankel transform</a></li> <li><a href="/facts/Fourier%E2%80%93Bessel_series/5DKL56kE">Fourier–Bessel series</a></li></ul> <ul><li><a href="/facts/Leonard_Carlitz/b21l2nBP">Carlitz, Leonard</a> (1957). "A Note on the Bessel Polynomials". <i>Duke Math. J</i>. 24 (2): 151–162. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1215%2FS0012-7094-57-02421-3">10.1215/S0012-7094-57-02421-3</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0085360">0085360</a>.</li> <li>Fakhri, H.; Chenaghlou, A. (2006). "Ladder operators and recursion relations for the associated Bessel polynomials". <i>Physics Letters A</i>. 358 (5–6): 345–353. <a href="/facts/Bibcode_(identifier)/9HtdQSGB">Bibcode</a>:<a href="https://ui.adsabs.harvard.edu/abs/2006PhLA..358..345F">2006PhLA..358..345F</a>. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1016%2Fj.physleta.2006.05.070">10.1016/j.physleta.2006.05.070</a>.</li> <li>Roman, S. (1984). <i>The Umbral Calculus (The Bessel Polynomials §4.1.7)</i>. New York: Academic Press. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-486-44139-9.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://www.encyclopediaofmath.org/index.php?title=Bessel_polynomials">"Bessel polynomials"</a>, <i><a href="/facts/Encyclopedia_of_Mathematics/WC6mGtPm">Encyclopedia of Mathematics</a></i>, <a href="/facts/European_Mathematical_Society/B3h7b672">EMS Press</a>, 2001 [1994]</li> <li><a href="/facts/Eric_W._Weisstein/FbMuVDep">Weisstein, Eric W.</a> <a href="https://mathworld.wolfram.com/BesselPolynomial.html">"Bessel Polynomial"</a>. <i><a href="/facts/MathWorld/yc1jodbq">MathWorld</a></i>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Krall, H. L.; Frink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65 (1): 100–115. doi:10.2307/1990516. <a href="https://doi.org/10.2307%2F1990516" target="_blank">https://doi.org/10.2307%2F1990516</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics). New York: Springer. ISBN 978-0-387-09104-4. <a href="978-0-387-09104-4" target="_blank">978-0-387-09104-4</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Berg, Christian; Vignat, Christophe (2008). "Linearization coefficients of Bessel polynomials and properties of Student-t distributions" (PDF). Constructive Approximation. 27: 15–32. doi:10.1007/s00365-006-0643-6. Retrieved 2006-08-16. <a href="http://www.math.ku.dk/~berg/manus/bessel.pdf" target="_blank">http://www.math.ku.dk/~berg/manus/bessel.pdf</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics). New York: Springer. ISBN 978-0-387-09104-4. <a href="978-0-387-09104-4" target="_blank">978-0-387-09104-4</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Wolfram Alpha example <a href="http://www.wolframalpha.com/input/?i=15x^3%2B15x^2%2B6x%2B1%3D%3DSqrt%5B2%2F%28Pi+x%29%5D+Exp%5B1%2Fx%5D+BesselK%5B3.5%2C+1%2Fx%5D" target="_blank">http://www.wolframalpha.com/input/?i=15x^3%2B15x^2%2B6x%2B1%3D%3DSqrt%5B2%2F%28Pi+x%29%5D+Exp%5B1%2Fx%5D+BesselK%5B3.5%2C+1%2Fx%5D</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Dita, Petre; Grama, Nicolae (May 14, 1997). "On Adomian's Decomposition Method for Solving Differential Equations". arXiv:solv-int/9705008. <a href="/wiki/ArXiv_(identifier)" target="_blank">/wiki/ArXiv_(identifier)</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics). New York: Springer. ISBN 978-0-387-09104-4. <a href="978-0-387-09104-4" target="_blank">978-0-387-09104-4</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Krall, H. L.; Frink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65 (1): 100–115. doi:10.2307/1990516. <a href="https://doi.org/10.2307%2F1990516" target="_blank">https://doi.org/10.2307%2F1990516</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Krall, H. L.; Frink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65 (1): 100–115. doi:10.2307/1990516. <a href="https://doi.org/10.2307%2F1990516" target="_blank">https://doi.org/10.2307%2F1990516</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Krall, H. L.; Frink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65 (1): 100–115. doi:10.2307/1990516. <a href="https://doi.org/10.2307%2F1990516" target="_blank">https://doi.org/10.2307%2F1990516</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Krall, H. L.; Frink, O. (1948). "A New Class of Orthogonal Polynomials: The Bessel Polynomials". Trans. Amer. Math. Soc. 65 (1): 100–115. doi:10.2307/1990516. <a href="https://doi.org/10.2307%2F1990516" target="_blank">https://doi.org/10.2307%2F1990516</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Wolfram, D.A. (2024). "Inverse connection formulae for generalised Bessel polynomials". Bulletin of the Australian Mathematical Society. 110 (3): 480–487. doi:10.1017/S0004972724000285. <a href="/wiki/Doi_(identifier)" target="_blank">/wiki/Doi_(identifier)</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Grosswald, E. (1978). Bessel Polynomials (Lecture Notes in Mathematics). New York: Springer. 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