A Selection of Problems from A.A. Markov’s Calculus of Probabilities: Calculating Probabilities of Repeated Independent Events, Part 2

Author(s):

Alan Levine (Franklin and Marshall College)

Turning to the calculation of the sum \(S\), we now show that we can successfully take advantage of an expansion as a continued fraction²⁶ which follows as a particular case of Gauss’ Formula:

\[\frac{F(\alpha, \beta+1, \gamma+1, x)}{F(\alpha, \beta, \gamma, x)} = \cfrac{1}{1-\cfrac{ax}{1-\cfrac{bx}{1-\cfrac{cx}{1-\cfrac{dx}{1-\ddots}}}}}\] where \(F(\alpha, \beta, \gamma, x)\) and \(F(\alpha, \beta + 1, \gamma + 1, x)\) denote the “hypergeometric” series:²⁷

\[ 1 + \frac{\alpha\cdot \beta}{1\cdot \gamma} x + \frac{\alpha(\alpha+1)\beta(\beta+1)}{1\cdot 2 \gamma (\gamma + 1)} x^2 + \cdots \] and \[ 1 + \frac{\alpha(\beta+1)}{1(\gamma+1)} x + \frac{\alpha(\alpha+1)(\beta+1)(\beta+2)}{1\cdot 2 (\gamma + 1)(\gamma + 2)} x^2 + \cdots \] and the coefficients \(a, b, c,d, \dots\) are determined by the equations
\[\begin{align} a & = \frac{\alpha (\gamma-\beta)}{\gamma(\gamma + 1)},&\ \ b & = \frac{(\beta+1)(\gamma+ 1 - \alpha)}{(\gamma+1)(\gamma+2)}, \\
c&= \frac{(\alpha+1)(\gamma+ 1 - \beta)}{(\gamma+2)(\gamma+3)}, &\ \ d &= \frac{(\beta+2)(\gamma+ 2 - \alpha)}{(\gamma+3)(\gamma+4)}, \dots\\
\end{align}\]

Referring to the result of Gauss’ Formula, we note that it follows from the following simple connections between different hypergeometric series:
\[F(\alpha, \beta+1, \gamma+1, x) - F(\alpha, \beta, \gamma, x) = ax F(\alpha+1, \beta+1, \gamma+2, x),\]
\[F(\alpha+1, \beta+1, \gamma+2, x) - F(\alpha, \beta+1, \gamma+1, x) = bx F(\alpha+1, \beta+2, \gamma+3, x),\]
\[F(\alpha+1, \beta+2, \gamma+3, x) - F(\alpha+1, \beta+1, \gamma+2, x) = cx F(\alpha+2, \beta+2, \gamma+4, x), \dots\]

To apply Gauss’ Formula to the expansion of \(S\) as a continued fraction, we should let \(\alpha = -n + l + 1\), \(\beta = 0\), \(\gamma = l+1\), \(x = -\frac{p}{q}\), which gives us the equation \[S = \cfrac{1}{1-\cfrac{c_1}{1+\cfrac{d_1}{1-\cfrac{c_2}{1+\cfrac{d_2}{1-\ddots}}}}},\] where, in general, \[c_k = \frac{(n-k-l)(l+k)}{(l+2k-1)(l+2k)}\frac{p}{q},\ \ \ \ d_k = \frac{k(n+k)}{(l+2k)(l+2k+1)}\frac{p}{q}.\]

We have here not an infinite, but a finite continued fraction, the last term of which will be \(\frac{d_{n-l-1}}{1}\) since \(c_{n-l} = 0\).

It is also not difficult to see that each of the numbers \(c_k\) is less than 1 only if \(\frac{n-l-1}{l+2}\frac{p}{q} <1\), as we will assume in the following argument.

Hence, denoting for brevity the continued fraction \[\cfrac{c_k}{1+\cfrac{d_k}{1-\cfrac{c_{k+1}}{1+\cfrac{d_{k+1}}{1-\ddots}}}}\] by the symbol \(\omega_k\), we have \(0<\omega_k< c_k\) and then we can establish a series of inequalities:
\[ S = \frac{1}{1-\omega_1} < \frac{1}{1-c_1}, \ \ S > \cfrac{1}{1-\cfrac{c_1}{1+\cfrac{d_1}{1-c_2}}},\]
\[S < \cfrac{1}{1-\cfrac{c_1}{1+\cfrac{d_1}{1-\cfrac{c_2}{1+\cfrac{d_2}{1-c_3}}}}}, \cdots\]

Continue to the third part of Markov’s analysis of the binomial distribution.

Skip to Suggestions for the Classroom and Conclusion.

[26] Markov's 1884 doctoral dissertation under the guidance of his advisor, P. L. Chebyshev was entitled, “On Some Applications of Algebraic Continued Fractions.” The concept of continued fractions dates back at least to John Wallis (1616–1703) in the 17th century.

More generally, a continued fraction is an expression of the form \[a_0 + \cfrac{b_1}{a_1 + \cfrac{b_2}{a_2 + \cfrac{b_3}{a_3 + \ddots}}}.\] The number of terms may be finite or infinite. If \(b_i = 1\), for all \(i\), the fraction is said to be “simple.” If all the terms of a finite continued fraction are rational, then the fraction represents a rational number. Infinite continued fractions represent irrational numbers.

[27] The hypergeometric series used here is generally denoted \[\,_2F_1(a,b;c,z) = \sum_{n=0}^\infty \frac{(a)_n(b)_n}{(c)_n} \frac{z^n}{n!},\] where \((k)_n = k (k+1)(k+2)\cdots(k+n-1) = \frac{(k+n-1)!}{(k-1)!}\) is the “ascending factorial” or “Pochhammer symbol.” As we have noted before, Markov did not use the symbol “\(!\).” These series have applications in many branches of mathematics, including differential equations and elliptic integrals. There are many identities involving hypergeometric series, some of which are used here. Although Euler and others used these series, the first comprehensive study was done by Gauss in 1813.

The subscripts 2 and 1 on \(F\) mean there are two ascending factorial expressions in the numerator and one in the denominator. This can be generalized to \(\,_p F_q\).

Many functions can be expressed as hypergeometric series; for example, \[\,_2 F_1 (1,1;2,-z) = \frac{\ln(1+z)}{z}.\]

Alan Levine (Franklin and Marshall College), "A Selection of Problems from A.A. Markov’s Calculus of Probabilities: Calculating Probabilities of Repeated Independent Events, Part 2," Convergence (November 2023)

Convergence

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History of Mathematics