A Short Proof of the Binomial Identities of Frisch and Klamkin

Ulrich Abel Department MND Technische Hochschule Mittelhessen Wilhelm-Leuschner-Straße 13 61169 Friedberg Germany

                    ulrich.abel@mnd.thm.de

Abstract We present short proofs for Frisch’s identity and Klamkin’s identity. Furthermore,

we deduce variants of Frisch’s and Klamkin’s identities involving infinite series.

1 Introduction

In their recent paper [4], Gould and Quaintance gave new proofs of Frisch’s identity

∑_{k=0}^{n} (−1)^{k} \binom{n}{k}\binom{b + k}{c}^{−1} = \frac{c}{n + c}\binom{n + b}{b − c}^{−1}  (b ≥ c > 0)  (1)

and Klamkin’s identity

∑_{k=0}^{n} \binom{n}{k}\binom{x}{k + b}^{−1} = \frac{x + 1}{x − n + 1}\binom{x − n}{b}^{−1}  (x − n ≥ b ≥ 0) .  (2)

As usual we put \binom{x}{0} = 1. For suitable real numbers x, y, the binomial coefficient \binom{x}{y} is to be read in terms of the gamma function, i.e., \binom{x+y}{y} = Γ (x + y + 1) / (Γ (x + 1) Γ (y + 1)). The proofs in [4] are based on the well-known formula of Gauss for the hypergeometric function 2F_{1} (a, b; c; 1).

In 1969, Ragnar Frisch (1895–1973) was awarded the first Nobel Prize in Economic Sciences. Identity (1) appeared in Frisch’s 1926 dissertation [2]. It was cited and proved in the 2nd edition 1927 of the book [5, pp. 337–338] by Netto. A further proof of Frisch’s identity (1), a two-page calculation involving an application of Melzak’s formula, can be found in the new book [6, Section 7.2].

In [4] the authors report that identity (2) in its original form [4, Eq. (1)] with x = n + a was stated by Murray S. Klamkin in a letter to Henry W. Gould on May 16, 1966. It is tabulated as Formula (4.2) in Gould’s collection [3]. Identity (4.6) in [3] is a special case of this.

The purpose of this note are elementary short proofs without application of hypergeometric functions. They are based on the use of the Euler beta function

B (x, y) = ∫_{0}^{1} t^{x−1} (1 − t)^{y−1} dt = \frac{Γ (x) Γ (y)}{Γ (x + y)} (x, y > 0) .

Furthermore, we present variants of Frisch’s and of Klamkin’s identities involving infinite series.

2 Proof of Frisch’s and Klamkin’s formulas

By direct calculation, we obtain, for n = 0, 1, 2, … and b ≥ c > 0,

∑_{k=0}^{n} (−1)^{k} \binom{n}{k}\binom{b + k}{c}^{−1} = c ∑_{k=0}^{n} (−1)^{k} \binom{n}{k}B (b − c + 1 + k, c)

= c ∑_{k=0}^{n} (−1)^{k} \binom{n}{k} ∫_{0}^{1} t^{b−c+k} (1 − t)^{c−1} dt

= c · B (b − c + 1, c + n) = \frac{c}{n + c}\binom{n + b}{b − c}^{−1},

which is Frisch’s identity (1). For suitable x ≠ −1, we have

\frac{1}{x + 1} ∑_{k=0}^{n} \binom{n}{k}\binom{x}{k + b}^{−1} = ∑_{k=0}^{n} \binom{n}{k}B (b + k + 1, x − b − k + 1)

= ∑_{k=0}^{n} \binom{n}{k} ∫_{0}^{1} t^{b} (1 − t)^{x−b} (\frac{t}{1 − t})^{k} dt

= B (b + 1, x − b − n + 1) = \frac{1}{x − n + 1}\binom{x − n}{b}^{−1},

which proves Klamkin’s identity (2).

3 Variants of Frisch’s and Klamkin’s identities

Since \binom{n}{k} = 0, for integers k > n ≥ 0, the left-hand sides in both identities (1), (2) can be written as infinite sums. Formally replacing n with −n in those equations and using the obvious binomial identity

(−1)^{k} \binom{−n}{k} = \binom{n + k − 1}{k}

yields

∑_{k=0}^{∞} \binom{n + k − 1}{k}\binom{b + k}{c}^{−1} = \frac{c}{c − n}\binom{−n + b}{b − c}^{−1},

∑_{k=0}^{∞} (−1)^{k} \binom{n + k − 1}{k}\binom{x}{k + b}^{−1} = \frac{x + 1}{x + n + 1}\binom{x + n}{b}^{−1}.

We shall show that both equations are valid also for positive integers n, if the parameters a, b, c are chosen in an appropriate manner.

First we deduce the variant of Frisch’s identity.

Theorem 1. For b ≥ c > n ≥ 1,

∑_{k=0}^{∞} \binom{n + k − 1}{k}\binom{b + k}{c}^{−1} = \frac{c}{c − n}\binom{b − n}{b − c}^{−1}.  (3)

The infinite series in Eq. (3) is convergent since

\binom{n + k − 1}{k}\binom{b + k}{c}^{−1} = Γ (n) \frac{Γ (n + k)}{Γ (1 + k)}Γ (c + 1) \frac{Γ (b − c + 1 + k)}{Γ (b + 1 + k)} ∼ \frac{Γ (n) Γ (c + 1)}{k^{1−n+c}}

as k → ∞ (see, e.g., [1, (6.1.46)]) and c > n.

Proof of Theorem 1. By using the well-known power series expansion

∑_{k=0}^{∞} \binom{n + k − 1}{k}z^{k} = (1 − z)^{−n} (|z| < 1)

instead of the binomial formula, it follows in the same manner as in the preceding section that

∑_{k=0}^{∞} \binom{n + k − 1}{k}B (x + k, y) = B (x, y − n) (x > 0, y > n) .

Rewritten in terms of binomial coefficients the latter identity takes the form

x ∑_{k=0}^{∞} \binom{n + k − 1}{k}\binom{x + y + k − 1}{y}^{−1} = y\binom{x + y − n − 1}{x}^{−1} (x > 0, y > n) .

Replacing x with b − c + 1 and y with c, we obtain

∑_{k=0}^{∞} \binom{n + k − 1}{k}\binom{b + k}{c}^{−1} = \frac{c}{b − c + 1}\binom{b − n}{b − c + 1}^{−1} (b > c − 1, c > n) .

Now Eq. (3) follows since

\frac{c}{b − c + 1}\binom{b − n}{b − c + 1}^{−1} = \frac{c}{c − n}\binom{b − n}{b − c}^{−1}.

This completes the proof. Remark 2. An alternative approach is the observation

∑_{k=0}^{∞} \binom{n + k − 1}{k}\binom{b + k}{c}^{−1} = \binom{b}{c}^{−1} _{2}F_{1} (1 + b − c, n; b + 1; 1) .

This formula immediately implies Eq. (3) since, by Gauss’s formula [1, (15.1.20)],

2F_{1} (1 + b − c, n; b + 1; 1) = \frac{Γ (c − n) Γ (b + 1)}{Γ (c) Γ (b − n + 1)} (c > n, b > −1) .

We close with the variant of Klamkin’s identity. Theorem 3. Let −a ∉ ℕ and −b ∉ ℕ. For −a − 1 > n ≥ 1 and a − b + 1 ∉ ℕ,

∑_{k=0}^{∞} (−1)^{k} \binom{n + k − 1}{k}\binom{a}{b + k}^{−1} = \frac{a + 1}{a + n + 1}\binom{a + n}{b}^{−1}.  (4)

The method of proof using the beta integral does not work in the case of Theorem 3 because the arising sum becomes divergent. Therefore, we use the Gauss formula as in the preceding remark. Proof of Theorem 3. Noting that

(−1)^{k} \binom{a}{b + k}^{−1} = \binom{a}{b}^{−1} \frac{Γ (b + 1 + k)}{Γ (b + 1)} \frac{Γ (b − a)}{Γ (b − a + k)}.

we have

(−1)^{k} \binom{n + k − 1}{k}\binom{a}{b + k}^{−1} = γ \frac{Γ (n + k)}{Γ (1 + k)} \frac{Γ (b + 1 + k)}{Γ (b − a + k)} ∼ γk^{n+a}

as k → ∞, where γ = Γ (n) \frac{Γ(b−a)}{Γ(b+1)} \binom{a}{b}^{−1}. If a + 1 + n < 0 we have

(−1)^{k} \binom{n + k − 1}{k}\binom{a}{b + k}^{−1} = O (k^{−η}) (k → ∞)

with a certain constant η > 1. This shows the convergence of the infinite series in Eq. (4). Finally, we observe

∑_{k=0}^{∞} (−1)^{k} \binom{n + k − 1}{k}\binom{a}{b + k}^{−1} = \binom{a}{b}^{−1} _{2}F_{1} (n, b + 1; b − a; 1) ,

for a + 1 + n < 0 and a − b + 1 ∉ ℕ. Using

2F_{1} (n, b + 1; b − a; 1) = \frac{Γ (−a − n − 1) Γ (b − a)}{Γ (b − a − n) Γ (−a − 1)} = \frac{Γ (a − b + n + 1) Γ (a + 2)}{Γ (a − b + 1) Γ (a + 2 + n)}

the desired formula Eq. (4) follows after a short calculation.

4 Acknowledgment

The author is grateful to the anonymous reviewer for a thorough reading of the manuscript and valuable comments.

References

[1] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Appl. Math. Ser. 55, National Bureau of Standards, 1972.

[2] R. Frisch, Sur les semi-invariants et moments employés dans l’étude des distributions statistiques, Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo, II, Historisk-Filosofisk Klasse 3 (1926), 1–87. Quoted by Th. Skolem, p. 337, in Netto’s Lehrbuch.

[3] H. W. Gould, Combinatorial Identities, Morgantown, West Virginia, 1972.

[4] H. W. Gould and J. Quaintance, On the binomial identities of Frisch and Klamkin, J. Integer Sequences 19 (2016) Article 16.7.7.

[5] E. Netto, Lehrbuch der Kombinatorik, 2nd edition, Chelsea Publications, 1927.

[6] J. Quaintance and H. W. Gould, Combinatorial Identities for Stirling Numbers, World Scientific Press, 2016.

2010 Mathematics Subject Classification: Primary 05A19. Keywords: combinatorial identity, binomial coefficient, beta function.

Received October 29 2019; revised version received March 30 2020. Published in Journal of Integer Sequences, June 12 2020.

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