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.

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[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

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

{\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

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 {\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
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