binomial expansions in factorial powers

Binomial Expansions in Factorial PowersAuthor(s): Louis BrandSource: The American Mathematical Monthly, Vol. 67, No. 10 (Dec., 1960), pp. 953-957Published by: Mathematical Association of AmericaStable URL: http://www.jstor.org/stable/2309220 .

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BINOMIAL EXPANSIONS IN FACTORIAL POWERS* LOUIS BRAND, University of Houston

1. Definitions. In the calculus of finite differences the factorial power x(n)

plays a role analogous to that of Xn in the differential calculus. Yet at present there is not only a wide diversity of notations for the factorial power [1], but also a disagreement as to its definition when the index is negative. Moreover the notation (ax+b)(n) is sometimes used in a way inconsistent with the definition of x(n) and this usage leads to ambiguities which may be a fertile source of error. The way out of this confusion is to use a single definition applicable to all real values of the index. Moreover the alleged factorial powers (ax+b)(n) are readily converted into genuine factorial powers.

When n is a positive integer the factorial power x() is defined as the product of n factors

(1) (n) = x(x- (x n + This definition leads to the functional equation

(A) X(m+n) = X(m)(X - = X(n)(X-n)(m)

for factorial powers with positive index. Just as the functional equation x18+n =xXmxn for ordinary powers is used to define the meaning of zero, negative, rational, and eventually all real exponents, we use (A) to obtain a definition of x() for all real n. We therefore postulate the truth of (A) for all real values of m and n.

When m = 0, (A) becomes X(n) = X()X(n); hence if x # 0,

(2) x() -1.

When m -n, we have x(?) x(-n) (x +n) (n) or x(-) =/(x +n) (n). Hence if n is a positive integer and xO0, we have from (1)

(3) X (n (x+ 1)(x+ 2) ... (x+ n)

Thus (1), (2) and (3) define factorial powers for all integral indices provided x ! 0. We shall see that we can dispense with this last proviso.

Next let x be any real number and consider the function f(x) = x(z). Letting m = 1, n = x-1 in (A) we have

f(x) = -x() = x(x - 1)(x-1) = xf(x - 1).

But the gamma function r(x) =f,,etx-ldt hasr(x+?) =xr(x) as functional equation. Hence when x is not a negative integer we define

* Presented to a joint meeting of the Mathematics and Physics Section of the Texas Academy of Sciences and the Texas Section of the Association, Austin, December 11, 1959.

953



954 BINOMIAL EXPANSIONS IN FACTORIAL POWERS [December

(4) X(x) = r(x + 1), x 5.! - 1 - 2

When x = 0 this gives 0() =r(1) = 1; we may therefore remove the restriction x#O in definitions (2) and (3); thus

(5) 0(0) = 1 and O = 1/n!

in sharp contrast with ordinary powers for which 0? is an indeterminate form and 0-1 has no meaning.

Again from (2) we have X() = X(n+xn) = X()( - -n) (x-); using (4), we Inow give the general definitions of the factorial power [2]:

(6) x(n) - r(x + 1) r(x+ 1-n)

for all real values of x and n for which the gamma functions exist. This definition comprises all cases if we agree that, when the gamma function is infinite (for arguments 0, -1, -2, * - * ), (7) X lim (x + E)(n)

f >0

Thus from (6), (-1)2=r(o)/r(-2) is not defined; but from (7)

(-1)(2) = lim (e - 1)(2) = lim (E)- = lim (e - 1)(E - 2) - 2! e- *0 eo r-+ 1 (E - 2) e6-40

in agreement with (1). An excellent occasion to use (6) arises in plotting the graph of the interesting

continuous curve y =0 (x) the 'sidewinder. " The basic functional equation (A) for the factorial power follows at once

from (6) and the identity:

r(x+ 1) _ r(x+ 1) r(x-m+ 1) r((x + 1-m-n) r(x+ 1-m) r(x-m+ 1-n)

By taking n =-m in (A) we get the important special case:

(9) X(m)(X - m)(m) = xm)(X + m) - 1.

2. Differences. The difference of a function f(x) is defined as Af(x) =f(x+ 1) -f(x). The prime importance of the factorial power is due to the formula

(B) A(x + c)(n) = n(x + c) (n)

and the corresponding antidifference

(C) A1(X+C)(n) =(x + c) (n+1), n+ 1

where co is an arbitrary periodic function of period 1. The proof of (B) follows



1960] BINOMIAL EXPANSIONS IN FACTORIAL POWERS 955

at once from (6) and the functional equation r (x + i) = xr (x):

A(x + C)(n) _ (x + C + 1)(n) - (x + 6)(n)

r(x+c+2) r(x+c+1)

r(x + c + 2-n) r(x + c + 1 -n)

r(x?c+1) ( x+c+ 1

r(x + c1 -) x + + 1 -n r(x + C + 1)_= (X + c) (

r(x+c+2 - n)

3. Diverse notations. We pass now to some matters of current usage relative to factorial powers.

First, a number of authors [3] define

X(=n) 1/{ x(x + 1) ... (x + n - 1)}.

Although this definition preserves the difference relation (B), it is inadvisable because it does not conform to the functional equation (A) and requires a separate definition for negative indices. Others [4] avoid negative indices alto- gether or introduce special notations, thus restricting the generality of (A) and the difference formula (B).

A more serious defect [5] is to write the factorial expression (which we de- note by an index in brackets)

(10) (ax+b)(n]=(ax+b){a(x-1) +b} . . .{a(x-n+ 1) +b}

as a factorial power; whereas, in conformity with (1),

(ax + b)(n) = (ax + b)(ax + b- 1) . . . (ax + b-n + 1).

However (ax + b) (n] is readily expressed as a factorial power:

(11) (ax + b)[nl = an(X + (b/a))(n);

its difference then follows from (10):

(12) A(ax + b) [n] = ann(x + (b/a)) (n-1) = atn(ax + b)El'].

4. Binomial expansions. By means of (6) the binomial coefficient

( x x(k) r(x + 1) k k(k) (x + 1-k)r(k + 1)

is generalized to all values of x and k. Moreover this generalization preserves the relations

(14) () (x) 1 (X) ( x) (x + 1)(x) ( x)



956 BINOMIAL EXPANSIONS IN FACTORIAL POWERS [December

originally suggested in the integral case by interpreting (D) as the number of combinations of x things taken k at a time.

The generalized figurate numbers F, defined by Frankel [6] are actually the coefficients in the expansion of (1 -x)-8:

(15) (1-x)= F , | X <1 k=O

Consequently for all integral values of n,

(16) Fb = (-~n 1)k (n ) _k(n) (k) (16) F

The definition (6) may now be used to define Fk for all real n. The "criss-cross multiplication" of two sequences ak * bk defined by Frankel in Section 5 is their Cauchy-product sequence; and the theorems in his Section 6 follow at once from this fact and equation (15). For example (1 -X)-m-n = (1 -x)-m(l -X)- yields Fk +n= Fk *Fk.

The binomial theorem may be extended to factorial powers. Thus when n is a positive integer

(17) ( ) ) E ( k=O k

a result known as Vandermonde's theorem. This identity, trivial for n =1, may be proved by induction.

A proof that reveals the essential connection of (17) with the binomial theorem for powers depends upon the identity (1 +t)x+y = (1 +t)z(l +t)y; for if we replace the powers of 1 +t by binomial series and equate the coefficients of tn on sides, we have

(x + Y ) (O) ( x)(Y) + ()(n)

On multiplying this equation by n! and using n! = ()(n-k)!k! we obtain (17). (See Frankel [6 ], Sec. 9.)

Under appropriate conditions the expansion (17) also applies when n is negative or fractional; in general, the series is then infinite and conditions for its convergence to f(x) must be investigated. But unlike the ordinary binomial expansion, the series for (x + y) (nwill terminate when y is a positive integer. For example,

2-\

(2+2)(1=E ) (1) (-1-k) 2 (7 k=O k

= (1) (-1)2(0) - (I) (-2)2(1) + (_) (-8)2 (2)

= 3 _f 1S +i-I 1% =- 2



1960] BINOMIAL EXPANSIONS IN FACTORIAL POWERS 957

which is U)(-1), Moreover,

Q + 2)(1'2) = E ( (1122-k)2 () k=O k

= (1) (112)2(0) + 4. ) (-)1/2)2(1) - j(1)(-312)2(2)

r(a( + i - )- 5

which is (5)(I) =P(7)/P(3). These examples suggest the

THEOREM. Whenever a binomial expansion in factorial powers terminates, it is correct irrespective of the index.

Proof. Let m be a positive integer; then from (A)

(X + m)(n) = (X + M)(m+n-m) = (X + M)(m)X(n-m)

= X(n-m)(X + m - n + n)(m)

-X(fl-m) X)(+ M - n)(mk)n(k) k=O k

Again from (A) we have

Xn-m(X + m - n) (rn-k) = X(n-m+m-k) = X(n-k)

hence

(x + m) ( k) = (ro k X(nk)n(k= k () nkM'k

in view of (13). This proves the theorem.

The author wishes to thank the referee for suggesting a perusal of and refer- ence to the paper of E. T. Frankel [6].

References 1. C. Jordan, Calculus of Finite Differences (2nd ed.), New York, 1947, Sec. 16. 2. C. Jordan, op. cit., p. 56. 3. Tomlinson Fort, Finite Differences and Difference Equations in the Real Domain, Oxford,

1948, p. 3. A. HIenry, Le Calcul de Differences Finies, Paris, 1932, pp. 30-31. N. E. Norlund, Vorlesungen ulber Differenzenrechnung, Berlin, 1924, p. 5.

4. C. H. Richardson, Introduction to the Calculus of Finite Differences, New York, 1954, p. 9. (But the unnecessary symbol xil- = (x-n+l)(") is introduced.) E. J. Cogan and R. Z. Norman, Handbook of Calculus, Difference and Differential Equations, Englewood Cliffs, N. J., 1958, p. 20.

5. C. H. Richardson, op. cit., p. 9; formula (6), p. 22. E. J. Cogan and R. Z. Norman, op. cit., Sec. 13.2, (2).

6. E. T. Frankel, A calculus of figurate numbers and finite differences, this MONTHLY, vol. 57, 1950, pp. 14-25.



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