Remodified Bessel Functions via Coincidences and Near Coincidences

Martin Griffiths School of Education University of Manchester M13 9PL United Kingdom

           martin.griffiths@manchester.ac.uk

Abstract

By considering a particular probabilistic scenario associated with coincidences, we are led to a family of functions akin to the modified Bessel function of the first kind. These are in turn solutions to a certain family of linear differential equations possessing structural similarities to the modified Bessel differential equation. The Stirling number triangle of the second kind arises quite naturally from these differential equations, as do more complicated, yet related, truncated number triangles, none of which appear in Sloane’s On-Line Encyclopedia of Integer Sequences.

1 Introduction

Let X ∼ Po(λ) denote a discrete random variable having the Poisson distribution [3] with parameter λ. The mass function of X is given by

P(X = k) = \frac{e^{−λ}λ^{k}}{k!} , k ≥ 0.

Furthermore, suppose that, for some n ∈ ℕ, the random variables X_{1}, X_{2}, …, X_{n} are independently and identically distributed as X. We shall term the event

C_{n} = ⋃_{k=0}^{∞}{X_{1} = X_{2} = ⋯ = X_{n} = k}

a coincidence. In order to place such an event into some of context, we might imagine n call centers each receiving an average of λ telephone calls per hour. In this situation a coincidence is said to occur when, in a given hour, all of the centers receive exactly the same number of calls.

Next, let r ∈ {1, 2, …, n} and S_{r} denote the set {1, 2, …, n}\{r}. We define the event A_{r} by

A_{r} = {X_{a_{1}} = X_{a_{2}} = ⋯ = X_{a_{n−1}} = k},

where {a_{1}, a_{2}, …, a_{n−1}} = S_{r}. The event

N_{n} = ⋃_{k=0}^{∞} ⋃_{r=1}^{n} {A_{r} ∩ {X_{r} = k + 1}}

is known as a near coincidence. Returning to the call-center scenario, a near coincidence is said to occur when, in a given hour, all but one of the centers receive exactly the same number of calls, with the remaining center receiving exactly one more call than all of the others.

In this paper we show how probabilities associated with coincidences, near coincidences and beyond, give rise to functions which may be regarded as extended versions of certain Bessel functions. Via the linear differential equations these functions satisfy, this leads first to the triangle of Stirling numbers of the second kind and then on to rather more complicated, yet related, truncated number triangles.

2 A connection with Bessel functions and Stirling numbers of the second kind

Bessel functions arise as solutions to certain linear differential equations. They come in several varieties, and we will be concerned here with a particular Bessel function that appears, amongst other things, in connection with special relativity [4, 5] and the Skellam distribution [10].

Definition 1. The modified Bessel function of the first kind [9],

I_{m}(x) = ∑_{k=0}^{∞} \frac{x^{2k+m}}{2^{2k+m}k!Γ(k + m + 1)},

is one of the solutions to the modified Bessel differential equation [8] given by

x^{2}y′′ + xy′ − (m^{2} + x^{2}) y = 0,

where Γ(x) is the gamma function [2, 6].

The probability of a coincidence occurring in any given hour is given by

P(C_{n}) = ∑_{k=0}^{∞} (\frac{e^{−λ}λ^{k}}{k!} )^{n}

= e^{−nλ} ∑_{k=0}^{∞} \frac{λ^{nk}}{(k!)^{n}} .

Then, noting that

e^{−2λ}I_{0}(2λ) = e^{−2λ} ∑_{k=0}^{∞} \frac{(2λ)^{2k}}{2^{2k}k!Γ(k + 1)}

= e^{−2λ} ∑_{k=0}^{∞} \frac{λ^{2k}}{(k!)^{2}}

= P(C_{2}),

we are led first to extend the definition of I_{0}(x) as follows: Definition 2. Let n ∈ ℕ. Then

I_{0}(n, x) = ∑_{k=0}^{∞} \frac{x^{nk}}{n^{nk}(k!)^{n}} .

The function I_{0}(n, x) is related to the probability P(C_{n}) by way of

P(C_{n}) = e^{−nλ} ∑_{k=0}^{∞} \frac{λ^{nk}}{(k!)^{n}}
= e^{−nλ}I_{0}(n, nλ).

It is clear that I_{0}(1, x) = e^{x} and I_{0}(2, x) = I_{0}(x). Note also that y = I_{0}(1, x) and y = I_{0}(2, x) satisfy y′ − y = 0 and xy′′ + y′ − xy = 0, respectively. We now find a third-order linear differential equation having I_{0}(3, x) as a solution. Result 3. The function y = I_{0}(3, x) is a solution to

x^{2}y′′′ + 3xy′′ + y′ − x^{2}y = 0.

Proof. Let y = I_{0}(3, x). We have

y′ = ∑_{k=1}^{∞} \frac{x^{3k−1}}{3^{3k−1}(k!)^{2}(k − 1)!}

and

y′′ = ∑_{k=1}^{∞} \frac{(3k − 1)x^{3k−2}}{3^{3k−1}(k!)^{2}(k − 1)!},

so that

x(y′ + xy′′) = ∑_{k=1}^{∞} \frac{x^{3k}}{3^{3k−2}k!((k − 1)!)^{2}} ,

and hence

(x(y′ + xy′′))′ = ∑_{k=1}^{∞} \frac{x^{3k−1}}{3^{3(k−1)}((k − 1)!)^{3}}

= ∑_{k=0}^{∞} \frac{x^{3k+2}}{3^{3k}(k!)^{3}}

= x^{2}y.

It follows from this that y = I_{0}(3, x) satisfies

x^{2}y′′′ + 3xy′′ + y′ − x^{2}y = 0,

as required.

Adopting a method similar to that used in Result 3, we may show that y = I_{0}(4, x) is a solution to

(x (x^{2}y′′′ + 3xy′′ + y′))′ − x^{3}y = 0,

and so on. It is in fact the case that I_{0}(n, x) satisfies the nth-order linear differential equation

∑_{k=1}^{n} S(n, k)x^{k−1}y^{(k)} − x^{n−1}y = 0,  (1)

where S(n, k) is a Stirling number of the second kind, enumerating the partitions of n distinct objects into exactly k non-empty parts, and y^{(k)} denotes the kth derivative of y with respect to x. The number triangle associated with S(n, k) appears in Sloane’s On-Line Encyclopedia of Integer Sequences [7] as sequence \underline{A008277}. It is straightforward to prove (1) by using induction in conjunction with the well-known result

S(n, k) = kS(n − 1, k) + S(n − 1, k − 1),  (2)

which may be found in [1] and [6].

3 Near coincidences

Let us now consider the probability of the occurrence of a near coincidence, assuming that n ≥ 2. We have

P(N_{n}) = \binom{n}{1} ∑_{k=0}^{∞} \frac{e^{−λ}λ^{k+1}}{(k + 1)!} (\frac{e^{−λ}λ^{k}}{k!} )^{n−1}

= ne^{−nλ} ∑_{k=0}^{∞} \frac{λ^{nk+1}}{(k!)^{n}(k + 1)}.

This leads us to generalize I_{0}(n, x) as follows:

Definition 4. For n ≥ m + 1,

I_{m}(n, x) = ∑_{k=0}^{∞} \frac{x^{nk+m}}{n^{nk+m}(k!)^{n}(k + 1)^{m}} .

Note then that P(N_{n}) = ne^{−nλ}I_{1}(n, nλ). In this section we will indeed consider the special case m = 1, which is the one associated with near coincidences.

Result 5. The function y = I_{1}(2, x) satisfies the differential equation

x^{3}y′′′ + 2x^{2}y′′ − xy′ + y − x^{2} (xy′ + y) = 0.

Proof. First,

(xI_{1}(2, x))′ = ∑_{k=0}^{∞} \frac{2(k + 1)x^{2k+1}}{2^{2k+1}(k!)^{2}(k + 1)}

= ∑_{k=0}^{∞} \frac{x^{2k+1}}{2^{2k}(k!)^{2}}

= xI_{0}(2, x).

Therefore

I_{0}(2, x) = \frac{1}{x} (xI′_{1}(2, x) + I_{1}(2, x))

= I′_{1}(2, x) + \frac{1}{x}I_{1}(2, x),

and so, since y = I_{0}(2, x) is a solution to xy′′ + y′ − xy = 0, we may obtain

x (I′_{1}(2, x) + \frac{1}{x}I_{1}(2, x))′′ + (I′_{1}(2, x) + \frac{1}{x}I_{1}(2, x))′ − x (I′_{1}(2, x) + \frac{1}{x}I_{1}(2, x)) = 0.

From this it follows that y = I_{1}(2, x) does in fact satisfy

x^{3}y′′′ + 2x^{2}y′′ − xy′ + y − x^{2} (xy′ + y) = 0.

Taking things further,

(x^{2}I_{1}(3, x))′ = ∑_{k=0}^{∞} \frac{3(k + 1)x^{3k+2}}{3^{3k+1}(k!)^{3}(k + 1)}

= ∑_{k=0}^{∞} \frac{x^{3k+2}}{3^{3k}(k!)^{3}}

= x^{2}I_{0}(3, x),

from which we may obtain, using Result 3, that y = I_{1}(3, x) is a solution to

x^{4}y′′′′ + 5x^{3}y′′′ + x^{2}y′′ + 2xy′ − 2y − x^{3} (xy′ + 2y) = 0.

More generally,

(x^{n−1}I_{1}(n, x))′ = ∑_{k=0}^{∞} \frac{n(k + 1)x^{nk+n−1}}{n^{nk+1}(k!)^{n}(k + 1)}

= ∑_{k=0}^{∞} \frac{x^{nk+n−1}}{n^{nk}(k!)^{n}}

= x^{n−1}I_{0}(n, x),

giving

I_{0}(n, x) = I′_{1}(n, x) + \frac{n − 1}{x}I_{1}(n, x).  (3)

Result 6. The function y = I_{1}(n, x) is a solution to

∑_{k=1}^{n} f(n, k)x^{k+1}y^{(k+1)} + (n − 1)(−1)^{n+1} (xy′ − y) − x^{n} (xy′ + (n − 1)y) = 0,  (4)

where

f(n, k) = S(n, k) + \frac{n − 1}{(k + 1)!} ∑_{j=k+1}^{n}(−1)^{j−k+1}j!S(n, j),  (5)

noting that the sum on the far right is defined to be zero when k ≥ n.

Proof. Starting with (3) and proceeding by induction gives

I_{0}^{(k)}(n, x) = I_{1}^{(k+1)}(n, x) + k!(n − 1) ∑_{j=0}^{k} \frac{(−1)^{k−j}I_{1}^{(j)}(n, x)}{x^{k−j+1}j!}.

From this we obtain, using (1) and induction once more, the general result that I_{1}(n, x) satisfies the differential equation

∑_{k=1}^{n} f(n, k)x^{k+1}y^{(k+1)} + (n − 1) ∑_{k=1}^{n}(−1)^{k+1}k!S(n, k) (xy′ − y)
− x^{n} (xy′ + (n − 1)y) = 0,  (6)

where f(n, k) is as given in the statement of the result.

Since, by definition,

∑_{k=1}^{n} S(n, k)(x)_{k} = x^{n},

where (x)_{k} = x(x − 1)(x − 2) ⋯ (x − k + 1) denotes the falling factorial, on setting x = −1 it follows that

(−1)^{n} = ∑_{k=1}^{n} S(n, k)(−1)_{k}

= ∑_{k=1}^{n} S(n, k)k!(−1)^{k}.

Therefore (6) may be simplified somewhat to give the desired result.

Since in Definition 4 we require n ≥ m + 1, the number triangle for f(n, k) is, unlike that for S(n, k), truncated. Its first few rows may be seen in Table 1 of Section 5.

4 Further coincidences

Next, let r, s ∈ {1, 2, …, n} such that r ≠ s, and let Q_{r,s} denote the set {1, 2, …, n}\{r, s}. We define B_{r,s} by

B_{r,s} = {X_{a_{1}} = X_{a_{2}} = ⋯ = X_{a_{n−2}} = k},

where {a_{1}, a_{2}, …, a_{n−2}} = Q_{r,s}, and consider the event

M_{n} = ⋃_{k=0}^{∞} ⋃_{r,s} {B_{r,s} ∩ {X_{r} = X_{s} = k + 1}} ,

where the inner union is over all possible pairs (r, s) such that r, s ∈ {1, 2, …, n} and r ≠ s. It is clear that

P(M_{n}) = \binom{n}{2} ∑_{k=0}^{∞} (\frac{e^{−λ}λ^{k+1}}{(k + 1)!} )^{2} (\frac{e^{−λ}λ^{k}}{k!} )^{n−2}

= e^{−nλ}\binom{n}{2} ∑_{k=0}^{∞} \frac{λ^{nk+2}}{(k!)^{n}(k + 1)^{2}}

= e^{−nλ}\binom{n}{2}I_{2}(n, nλ).

For this new scenario we now obtain, in correspondence to Result 6, a family of linear differential equations and the associated truncated number triangle.

First,

(x (x^{n−2}I_{2}(n, x))′)′ = (x ∑_{k=0}^{∞} \frac{n(k + 1)x^{nk+n−1}}{n^{nk+2}(k!)^{n}(k + 1)^{2}} )′

= (∑_{k=0}^{∞} \frac{x^{nk+n}}{n^{nk+1}(k!)^{n}(k + 1)})′

= ∑_{k=0}^{∞} \frac{n(k + 1)x^{nk+n−1}}{n^{nk+1}(k!)^{n}(k + 1)}

= ∑_{k=0}^{∞} \frac{x^{nk+n−1}}{n^{nk}(k!)^{n}}

= x^{n−1}I_{0}(n, x),

from which we have

I_{0}(n, x) = I′′_{2}(n, x) + \frac{2n − 3}{x}I′_{2}(n, x) + \frac{(n − 2)^{2}}{x^{2}} I_{2}(n, x).  (7)

It follows from this that

I_{0}^{(k)}(n, x) =I_{2}^{(k+2)}(n, x) + k!(2n − 3) ∑_{j=0}^{k} \frac{(−1)^{k−j}I_{2}^{(j+1)}(n, x)}{x^{k−j+1}j!}

+ k!(n − 2)^{2} ∑_{j=0}^{k} \frac{(−1)^{k−j}(k − j + 1)I_{2}^{(j)}(n, x)}{x^{k−j+2}j!}.

Then, to obtain a linear differential equation satisfied by I_{2}(n, x), we may use the fact that I_{0}(n, x) is a solution to (1) to give

∑_{k=1}^{n} g(n, k)x^{k+2}y^{(k+2)} + (n − 2)^{2} ∑_{k=1}^{n} k!S(n, k) ∑_{j=0}^{2} \frac{(−1)^{k+j}}{j!} x^{j}y^{(j)}(k − j + 1)
+ (2n − 3)(−1)^{n+1} (x^{2}y′′ − xy′) − x^{n} (x^{2}y′′ + (2n − 3)xy′ + (n − 2)^{2}y) = 0, (8)

where

g(n, k) =S(n, k) + \frac{2n − 3}{(k + 1)!} ∑_{j=k+1}^{n}(−1)^{j−k−1}j!S(n, j)

+ \frac{(n − 2)^{2}}{(k + 2)!} ∑_{j=k+2}^{n}(−1)^{j−k}j!S(n, j)(j − k − 1),  (9)

and the first and second sums on the right are defined to be zero when k ≥ n and k ≥ n − 1, respectively. The truncated number triangle for g(n, k) is shown in Table 2 of Section 5.

The sequence of numbers along the mth diagonal of the triangle of Stirling numbers of the second kind is given by {S(n + m, n) : n = 1, 2, 3, …}, where the convention is that the upper-most diagonal, consisting solely of 1s, is the zeroth diagonal. The sequences for m = 1, 2, 3 and 4 appear in [7] as \underline{A000127}, \underline{A001296}, \underline{A001297}, and \underline{A001298}, respectively. Using the recurrence relation (2), it is possible to show, by induction, that for each m ∈ ℕ there exists a polynomial p_{m}(x) such that S(n + m, n) = p_{m}(n), n = 1, 2, 3, . . .. It is in fact reasonably straightforward to show that p_{m}(x) has degree 2m and leading coefficient \frac{1}{2^{m}m!}. It follows from this, in conjunction with (5), that the mth diagonal of the table for f(n, k) is a polynomial sequence of degree 2m. On using (9), a similar result applies to the table for g(n, k).

It is possible to generalize the results (3) and (7). We have

x^{n−1}I_{0}(n, x) = (x ⋯ (x (x^{n−m}I_{m}(n, x))′)′ ⋯)′ ,

where the nesting is to a depth of m. It follows from this that

x^{n−1}I_{0}(n, x) = ∑_{k=1}^{m} S(m, k)x^{k−1} (x^{n−m}I_{m}(n, x))^{(k)} .  (10)

It is also straightforward to show that

(x^{q}h(x))^{(k)} = ∑_{j=0}^{k} x^{q−j}(q)_{j}\binom{k}{j}h^{(k−j)}(x),  (11)

where (q)_{0} = 1 by definition. From (10) and (11) we may obtain

I_{0}(n, x) = \frac{1}{x^{n−1}} ∑_{k=1}^{m} S(m, k)x^{k−1} ∑_{j=0}^{k} x^{n−m−j}(n − m)_{j}\binom{k}{j}I_{m}^{(k−j)}(n, x)

= ∑_{k=1}^{m} S(m, k) ∑_{j=0}^{k} x^{k−j−m}(n − m)_{j}\binom{k}{j}I_{m}^{(k−j)}(n, x).

This result, in conjunction with (1), allows us to find a linear differential equation satisfied by y = I_{m}(n, x) for any m ∈ ℕ.

5 Tables

n f(n, 1) f(n, 2) f(n, 3) f(n, 4) f(n, 5) f(n, 6) f(n, 7) f(n, 8)
2  2  1
3  1  5  1
4  4  13  9  1
5  1  35  45  14  1
6  6  81  190  110  20  1
7  1  189  721  686  224  27  1
8  8  421  2583 3759 1932  406  35  1

Table 1: The coefficients f(n, k) of the differential equation (4).

n g(n, 1) g(n, 2) g(n, 3) g(n, 4) g(n, 5) g(n, 6) g(n, 7) g(n, 8)
3  2  6  1
4 −2  21  11  1
5 46  50  69  17  1
6 −150 201  318  162  24  1
7 526  294  1421 1141  319  32  1
8 −1498 1429 5481 7035 3120  562  41  1

Table 2: The coefficients g(n, k) of the differential equation (8).

6 Acknowledgement

The author is grateful to the anonymous referee for providing a number of constructive comments and suggestions as to how the original version of this article might be improved.

References

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[5] B. H. Lavenda, Special relativity via modified Bessel functions, Zeitschrift für Naturforschung A 55a (2000), 745–753.

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[7] N. J. A. Sloane (Ed.), The On-Line Encyclopedia of Integer Sequences, 2011. Available

   at http://oeis.org/ .

[8] E. W. Weisstein, Modified Bessel differential equation, MathWorld, 2011. Available at

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[9] E. W. Weisstein, Modified Bessel function of the first kind, MathWorld, 2011. Available at

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2010 Mathematics Subject Classification: Primary 33C10; Secondary 11B73, 33C90, 34A05, 60E05. Keywords: Bessel functions, linear differential equations, Poisson distributions, Stirling numbers of the second kind.

(Concerned with sequences \underline{A000127}, \underline{A001296}, \underline{A001297}, \underline{A001298}, and \underline{A008277}.)

Received January 28 2011; revised version received May 29 2011. Published in Journal of Integer Sequences, June 11 2011.

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