Inmathematical analysis, theBessel–Clifford function, named afterFriedrich Bessel andWilliam Kingdon Clifford, is anentire function of twocomplex variables that can be used to provide an alternative development of the theory ofBessel functions. If

${\displaystyle \pi (x)=\frac{1}{\mathrm{\Pi }(x)}=\frac{1}{\mathrm{\Gamma }(x+1)}}$

is the entire function defined by means of thereciprocal gamma function, then the Bessel–Clifford function is defined by the series

${\displaystyle {\mathcal{C}}_n(z)=\sum _{k=0}^{\mathrm{\infty }}\pi (k+n)\frac{z^k}{k!}}$

The ratio of successive terms isz/k(n + k), which for all values ofz andn tends to zero with increasing k. By theratio test, this series converges absolutely for allz and n, and uniformly for all regions with bounded |z|, and hence the Bessel–Clifford function is an entire function of the two complex variablesn and z.

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

${\displaystyle x{y}^{″}+(n+1)y^{\prime }=y.}$

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

${\displaystyle {\mathcal{C}}_n(z)=\pi (n){}_0{F}_1(;n+1;z).}$

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

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

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

whenn 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 for₀F₁. We have, as a special case ofGauss's continued fraction

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

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

The Bessel–Clifford differential equation

${\displaystyle x{y}^{″}+(n+1)y^{\prime }=y}$

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

${\displaystyle {\mathcal{K}}_n(x)=\frac{1}{2}{\int }_0^{\mathrm{\infty }}\exp \left(-t-\frac{x}{t}\right)\frac{dt}{t^{n+1}}}$

which converges for${\displaystyle \mathrm{\Re }(x)>0}$, 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 ofK, 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 (whent is not zero) an absolutely convergent series for exp(t + z/t). Collecting terms int, we find on comparison with thepower series definition for${\displaystyle {\mathcal{C}}_n}$ that we have

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

This generating function can then be used to obtain further formulas, in particular we may useCauchy's integral formula and obtain${\displaystyle {\mathcal{C}}_n}$ for integern as

${\displaystyle {\mathcal{C}}_n(z)=\frac{1}{2\pi i}{\oint }_C\frac{\exp (t+z/t)}{t^{n+1}}dt=\frac{1}{2\pi }{\int }_0^{2\pi }\exp (z\exp (-i\theta )+\exp (i\theta )-ni\theta )d\theta .}$

This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(August 2009) (Learn how and when to remove this message)

• 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.

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