A note on binomial theorem.

*(English)*Zbl 1270.11015In this note the authors recall the well-known Binomial Theorem illustrated by [M. Abramowitz (ed.) and I. A. Stegun (ed.), Handbook of mathematical functions with formulas, graphs, and mathematical tables. New York etc.: Wiley (1972; Zbl 0543.33001)]:
\[
(x+a)^n = \sum_{k=0}^{n} {}^nC_k x^k a^{n-k}
\]
where \( ^nC_k \) is a binomial coefficient and \(n\) is a positive integer; the formula still holds for negative integer -\(n\) which converges for \(|x|<a\).

They also recall the more general form supplied by [L. G. Ronald, D. E. Knuth and O. Patashnik, Concrete mathematics: a foundation for computer science. 2nd ed. Amsterdam: Addison-Wesley (1994; Zbl 0836.00001)] as follows: \[ (x+a)^r = \sum_{k=0}^{\infty} {}^rC_k x^k a^{r-k} \] where \( ^rC_k \) is a binomial coefficient and \(r\) is a real number.

The authors introduce generalised binomial coefficients by using \(\Gamma\)-functions as follows: \[ ^{n+\alpha}G_r = \frac{\Gamma(n+\alpha+1)}{\Gamma(n+\alpha+1-r)r!} \] where \(n\) is an integer, \(k=0,1,2,\ldots \) and \(\alpha\) is a nonzero real number satisfying \(|\alpha|< 1\); the authors remark that \( ^{n+\alpha}G_r \) becomes \( ^nC_r \) for \(\alpha = 0\).

These newly defined coefficients have the following notable property \[ ^{n+\alpha}G_k = \frac{(n+\alpha)(n+\alpha-1) \cdots (n+\alpha-k+1)}{k!} \] that for \( k = n+\alpha \) means \( ^{n+\alpha}G_{n+\alpha} = 1 \), while for \(0 \leq k \leq 2\) becomes, respectively, \( ^{n+\alpha}G_0 = 1\), \( ^{n+\alpha}G_1 = n+\alpha\), \( ^{n+\alpha}G_2 = \frac{(n+\alpha)(n+\alpha-1)}{2!}\).

Further interesting properties, such as \( ^{n+\alpha}G_0 + ^{n+\alpha}G_1 + ^{n+\alpha}G_2 + \cdots = 2^{n+\alpha} \) and more complex ones, are also explored. The authors clarify that the introduction of Gamma function helps to find easily the coefficients for negative numbers (except negative integers) and they give some evaluations of \( ^{n+\alpha}G_r \) for non-integer real numbers.

The main result of the paper is the following Binomial Theorem for all real numbers: \[ (1+x)^{n+\alpha} = \sum_{k=0}^{\infty} {}^{n+\alpha}G_k x^k (|x|< 1) \] generalizing the similar version reported by Eric W. Weisstein \[ (1+x)^r = \sum_{k=0}^{\infty} {}^rC_k x^k \] in [“Binomial Theorem”, http://mathworld.wolfram.com/BinomialTheorem.html] as supplied for the first time by George B. Arfken in 1985.

Beyond basic differentiability rules easily available, \(e.g.\), in [Lars V. Ahlfors, Complex analysis. 2nd ed. Maidenhead, Berkshire: McGraw-Hill (1966; Zbl 0154.31904)], in the proof the authors employ the Maclaurin series pointing out that such method was already used by K. Ward [“Series Binomial Theorem”, http://www.trans4mind.com/].

They also recall the more general form supplied by [L. G. Ronald, D. E. Knuth and O. Patashnik, Concrete mathematics: a foundation for computer science. 2nd ed. Amsterdam: Addison-Wesley (1994; Zbl 0836.00001)] as follows: \[ (x+a)^r = \sum_{k=0}^{\infty} {}^rC_k x^k a^{r-k} \] where \( ^rC_k \) is a binomial coefficient and \(r\) is a real number.

The authors introduce generalised binomial coefficients by using \(\Gamma\)-functions as follows: \[ ^{n+\alpha}G_r = \frac{\Gamma(n+\alpha+1)}{\Gamma(n+\alpha+1-r)r!} \] where \(n\) is an integer, \(k=0,1,2,\ldots \) and \(\alpha\) is a nonzero real number satisfying \(|\alpha|< 1\); the authors remark that \( ^{n+\alpha}G_r \) becomes \( ^nC_r \) for \(\alpha = 0\).

These newly defined coefficients have the following notable property \[ ^{n+\alpha}G_k = \frac{(n+\alpha)(n+\alpha-1) \cdots (n+\alpha-k+1)}{k!} \] that for \( k = n+\alpha \) means \( ^{n+\alpha}G_{n+\alpha} = 1 \), while for \(0 \leq k \leq 2\) becomes, respectively, \( ^{n+\alpha}G_0 = 1\), \( ^{n+\alpha}G_1 = n+\alpha\), \( ^{n+\alpha}G_2 = \frac{(n+\alpha)(n+\alpha-1)}{2!}\).

Further interesting properties, such as \( ^{n+\alpha}G_0 + ^{n+\alpha}G_1 + ^{n+\alpha}G_2 + \cdots = 2^{n+\alpha} \) and more complex ones, are also explored. The authors clarify that the introduction of Gamma function helps to find easily the coefficients for negative numbers (except negative integers) and they give some evaluations of \( ^{n+\alpha}G_r \) for non-integer real numbers.

The main result of the paper is the following Binomial Theorem for all real numbers: \[ (1+x)^{n+\alpha} = \sum_{k=0}^{\infty} {}^{n+\alpha}G_k x^k (|x|< 1) \] generalizing the similar version reported by Eric W. Weisstein \[ (1+x)^r = \sum_{k=0}^{\infty} {}^rC_k x^k \] in [“Binomial Theorem”, http://mathworld.wolfram.com/BinomialTheorem.html] as supplied for the first time by George B. Arfken in 1985.

Beyond basic differentiability rules easily available, \(e.g.\), in [Lars V. Ahlfors, Complex analysis. 2nd ed. Maidenhead, Berkshire: McGraw-Hill (1966; Zbl 0154.31904)], in the proof the authors employ the Maclaurin series pointing out that such method was already used by K. Ward [“Series Binomial Theorem”, http://www.trans4mind.com/].

Reviewer: Enzo Bonacci (Latina)