Bessel-Clifford function

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In mathematical analysis, the Bessel-Clifford function is an entire function of two complex variables which can be used to provide an alternative development of the theory of Bessel functions. If

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

is the entire function defined by means of the reciprocal of the Gamma function, then the Bessel-Clifford function is defined by the series

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

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

1 Differential equation of the Bessel-Clifford function
2 Relation to Bessel functions
3 Recurrence relationship
4 Continued fraction
5 The Bessel-Clifford function of the second kind
6 Generating function
7 References

[edit] Differential equation of the Bessel-Clifford function

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

x y'' + (n + 1) y' = y .

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

${\mathcal C}_n(z) = \pi(n)\ _0F_1(n+1; z).$

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.

[edit] Relation to Bessel functions

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

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

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

$I_n(z) = (\frac{z}{2})^n {\mathcal C}_n(\frac{z^2}{4}).$

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

${\mathcal C}_n(z) = z^{-n/2} I_n(2 \sqrt{z});$

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

[edit] Recurrence relationship

From the defining series, it follows immediately that $\frac{d}{dx}{\mathcal C}_n(x) = {\mathcal C}_{n+1}(x).$ Using this, we may rewrite the differential equation for ${\mathcal C}$ as

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

which defines the recurrence relationship for the Bessel-Clifford function.

[edit] Continued fraction

From the recurrence relationship, we have, on dividing by ${\mathcal C}_{n+1}(x)$ , that

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

If we set $Q_n(x) = \frac{{\mathcal C}_{n+1}(x)}{{\mathcal C}_n(x)}$ , then this may be written

$Q_n(x) = \frac{1}{n+1+xQ_{n+1}(x)},$

which if iterated leads to

$Q_n(x) = \frac{1}{n+1 + \frac{x}{n+2+\frac{x}{n+3+ \cdots}}}.$

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

It follows directly from the series definition that ${\mathcal C}_{1/2}(x) = \frac{\sinh(2 \sqrt{x})}{\sqrt{\pi x}}$ and ${\mathcal C}_{-1/2}(x) = \frac{\cosh(2 \sqrt{x})}{\sqrt{\pi}},$ so that $Q_{-1/2}(x) = \frac{\tanh (2 \sqrt{x})}{\sqrt{x}}.$ From this, one can deduce both

$\tanh(x) = \frac{x}{1+\frac{x^2}{3+\frac{x^2}{5+\cdots}}}$

and

$\tan(x) = \frac{x}{1-\frac{x^2}{3-\frac{x^2}{5-\cdots}}}.$

The first formula is due to Gauss, and immediately demonstrates that e is irrational. The second formula is due to Lambert, and was used by both him and Legendre to prove that π is irrational. Legendre defined the Clifford-Bessel function in the process of deriving this continued fraction.

[edit] The Bessel-Clifford function of the second kind

The Bessel-Clifford differential equation

x y'' + (n + 1) y' = y

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

If we set

${\mathcal K}_n(x) = \frac{1}{2} \int_0^\infty \exp(-t-\frac{x}{t}) \frac{dt}{t^{n+1}}$

which converges for $\Re(x) > 0$ , 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 ${\mathcal K}$ correspond to the Bessel functions of the second kind. We have

$K_n(x) = (\frac{x}{2})^n {\mathcal K}_n(\frac{x^2}{4}).$

and

$Y_n(x) = (\frac{x}{2})^n {\mathcal K}_n(-\frac{x^2}{4}).$

In terms of K, we have

${\mathcal K}_n(x) = x^{-n/2} K_n(2 \sqrt{x}).$

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

[edit] Generating function

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 ${\mathcal C}_n$ that we have

$\exp(t + \frac{z}{t}) = \sum_{n=-\infty}^\infty t^n {\mathcal C}_n(z).$

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

${\mathcal C}_n(z) = \frac{1}{2 \pi i} \oint_C \frac{\exp(z+z/t)}{t^{n+1}} dt = \frac{1}{2 \pi}\int_0^{2 \pi} \exp(z(1+\exp(-i\theta))-ni\theta))d\theta.$

[edit] References

William Kingdon Clifford, On Bessel's Functions, Mathematical Papers, London (1882), pp 346-349
A. George Greenhill, The Bessel-Clifford function, and its applications, Philosophical Magazine, Sixth Series, (1919), pp 501-528
Adrien-Marie Legendre, Éléments de Géometrie, Note IV, (1802), Paris
Ludwig Schläfli, 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, (1868) pp 232-242
G. N. Watson, A Treatise on the Theory of Bessel Functions, Second Edition, Cambridge University Press
Rolf Wallisser, "On Lambert's proof of the irrationality of π", in Algebraic Number Theory and Diophantine Analysis, Franz Halter-Koch and Robert F. Tichy, (2000), Walter de Gruyer