Bessel–Clifford function

It follows from the above series on differentiating with respect to x that ${\displaystyle {\mathcal {C}}_{n}(x)}$ satisfies the linear second-order homogeneous differential equation

${\displaystyle xy''+(n+1)y'=y.\qquad }$

This equation is of generalized hypergeometric type, and in fact the Bessel–Clifford function is up to a scaling factor a Pochhammer–Barnes hypergeometric function; we have

${\displaystyle {\mathcal {C}}_{n}(z)=\pi (n)\ _{0}F_{1}(;n+1;z).}$

Unless n is a negative integer, in which case the right hand side is undefined, the two definitions are essentially equivalent; the hypergeometric function being normalized so that its value at z = 0 is one.

The Bessel function of the first kind can be defined in terms of the Bessel–Clifford function as

${\displaystyle J_{n}(z)=\left({\frac {z}{2}}\right)^{n}{\mathcal {C}}_{n}\left(-{\frac {z^{2}}{4}}\right);}$

when n is not an integer we can see from this that the Bessel function is not entire. Similarly, the modified Bessel function of the first kind can be defined as

${\displaystyle I_{n}(z)=\left({\frac {z}{2}}\right)^{n}{\mathcal {C}}_{n}\left({\frac {z^{2}}{4}}\right).}$

The procedure can of course be reversed, so that we may define the Bessel–Clifford function as

${\displaystyle {\mathcal {C}}_{n}(z)=z^{-n/2}I_{n}(2{\sqrt {z}});}$

but from this starting point we would then need to show ${\displaystyle {\mathcal {C}}}$ was entire.

From the defining series, it follows immediately that ${\displaystyle {\frac {d}{dx}}{\mathcal {C}}_{n}(x)={\mathcal {C}}_{n+1}(x).}$

Using this, we may rewrite the differential equation for ${\displaystyle {\mathcal {C}}}$ as

${\displaystyle x{\mathcal {C}}_{n+2}(x)+(n+1){\mathcal {C}}_{n+1}(x)={\mathcal {C}}_{n}(x),}$

which defines the recurrence relationship for the Bessel–Clifford function. This is equivalent to a similar relation for ₀F₁. We have, as a special case of Gauss's continued fraction

${\displaystyle {\frac {{\mathcal {C}}_{n+1}(x)}{{\mathcal {C}}_{n}(x)}}={\cfrac {1}{n+1+{\cfrac {x}{n+2+{\cfrac {x}{n+3+{\cfrac {x}{\ddots }}}}}}}}.}$

It can be shown that this continued fraction converges in all cases.

The Bessel–Clifford differential equation

${\displaystyle xy''+(n+1)y'=y\qquad }$

has two linearly independent solutions. Since the origin is a regular singular point of the differential equation, and since ${\displaystyle {\mathcal {C}}}$ is entire, the second solution must be singular at the origin.

If we set

${\displaystyle {\mathcal {K}}_{n}(x)={\frac {1}{2}}\int _{0}^{\infty }\exp \left(-t-{\frac {x}{t}}\right){\frac {dt}{t^{n+1}}}}$

which converges for ${\displaystyle \Re (x)>0}$, and analytically continue it, we obtain a second linearly independent solution to the differential equation.

The factor of 1/2 is inserted in order to make ${\displaystyle {\mathcal {K}}}$ correspond to the Bessel functions of the second kind. We have

${\displaystyle K_{n}(x)=\left({\frac {x}{2}}\right)^{n}{\mathcal {K}}_{n}\left({\frac {x^{2}}{4}}\right).}$

and

${\displaystyle Y_{n}(x)=\left({\frac {x}{2}}\right)^{n}{\mathcal {K}}_{n}\left(-{\frac {x^{2}}{4}}\right).}$

In terms of K, we have

${\displaystyle {\mathcal {K}}_{n}(x)=x^{-n/2}K_{n}(2{\sqrt {x}}).}$

Hence just as the Bessel function and modified Bessel function of the first kind can both be expressed in terms of ${\displaystyle {\mathcal {C}}}$, those of the second kind can both be expressed in terms of ${\displaystyle {\mathcal {K}}}$.

If we multiply the absolutely convergent series for exp(t) and exp(z/t) together, we get (when t is not zero) an absolutely convergent series for exp(t + z/t). Collecting terms in t, we find on comparison with the power series definition for ${\displaystyle {\mathcal {C}}_{n}}$ that we have

${\displaystyle \exp \left(t+{\frac {z}{t}}\right)=\sum _{n=-\infty }^{\infty }t^{n}{\mathcal {C}}_{n}(z).}$

This generating function can then be used to obtain further formulas, in particular we may use Cauchy's integral formula and obtain ${\displaystyle {\mathcal {C}}_{n}}$ for integer n as

${\displaystyle {\mathcal {C}}_{n}(z)={\frac {1}{2\pi i}}\oint _{C}{\frac {\exp(t+z/t)}{t^{n+1}}}\,dt={\frac {1}{2\pi }}\int _{0}^{2\pi }\exp(z\exp(-i\theta )+\exp(i\theta )-ni\theta )\,d\theta .}$

• Clifford, William Kingdon (1882), "On Bessel's Functions", Mathematical Papers, London: 346–349.
• Greenhill, A. George (1919), "The Bessel–Clifford function, and its applications", Philosophical Magazine, Sixth Series: 501–528.
• Legendre, Adrien-Marie (1802), Éléments de Géometrie, Note IV, Paris.
• Schläfli, Ludwig (1868), "Sulla relazioni tra diversi integrali definiti che giovano ad esprimere la soluzione generale della equazzione di Riccati", Annali di Matematica Pura ed Applicata, 2 (I): 232–242.
• Watson, G. N. (1944), A Treatise on the Theory of Bessel Functions (Second ed.), Cambridge: Cambridge University Press.
• Wallisser, Rolf (2000), "On Lambert's proof of the irrationality of π", in Halter-Koch, Franz; Tichy, Robert F. (eds.), Algebraic Number Theory and Diophantine Analysis, Berlin: Walter de Gruyer, ISBN 3-11-016304-7.