Falling and rising factorial

In mathematics, the falling or rising factorial denotes a function similar to exponentiation , but in which the factors fall or rise gradually, i.e. i.e., be reduced or increased by one.

definition

For natural numbers and with the -th decreasing or increasing factorial is noted as or (in some older textbooks also or ) and is defined as follows: ${\ displaystyle n}$${\ displaystyle k}$${\ displaystyle n \ geq k \ geq 0}$${\ displaystyle k}$${\ displaystyle n ^ {\ underline {k}}}$${\ displaystyle n ^ {\ overline {k}}}$${\ displaystyle (n) _ {k}}$${\ displaystyle n ^ {(k)}}$

${\ displaystyle n ^ {\ underline {k}}: = n (n-1) (n-2) \ cdots (n-k + 1) = {\ frac {n!} {(nk)!}}}$
${\ displaystyle n ^ {\ overline {k}}: = n (n + 1) (n + 2) \ cdots (n + k-1) = {\ frac {(n + k-1)!} {( n-1)!}}}$

Combinatorial interpretation

In the urn model , the falling factorial can be interpreted as the number of possibilities to remove balls from an urn with different balls , without replacing them, taking into account the sequence. There are candidates for the first ball , for the second ... and finally for the last ball . There are therefore options for the overall selection . ${\ displaystyle n}$${\ displaystyle k}$${\ displaystyle n}$${\ displaystyle n-1}$${\ displaystyle n-k + 1}$${\ displaystyle n (n-1) \ cdots (n-k + 1) = n ^ {\ underline {k}}}$

In general, the number of - permutations of a - set or, alternatively, the number of injective mappings of a - set into a - set. ${\ displaystyle n ^ {\ underline {k}}}$${\ displaystyle k}$${\ displaystyle n}$${\ displaystyle k}$${\ displaystyle n}$

generalization

The definition is analogous for a complex number and a natural number : ${\ displaystyle x}$${\ displaystyle k}$

${\ displaystyle x ^ {\ underline {k}}: = x (x-1) (x-2) \ cdots (x-k + 1)}$
${\ displaystyle x ^ {\ overline {k}}: = x (x + 1) (x + 2) \ cdots (x + k-1)}$

One can then understand and as complex polynomials in . ${\ displaystyle x ^ {\ underline {k}}}$${\ displaystyle x ^ {\ overline {k}}}$${\ displaystyle x}$

For the increasing factorial corresponds to the Pochhammer symbol . ${\ displaystyle x \ in \ mathbb {N}}$${\ displaystyle x ^ {\ overline {k}}}$ ${\ displaystyle (x, k)}$

properties

Calculation rules

The following calculation rules apply:

${\ displaystyle x ^ {\ underline {1}} = x ^ {\ overline {1}} = x}$
${\ displaystyle x ^ {\ underline {0}} = x ^ {\ overline {0}} = 1}$
${\ displaystyle (-x) ^ {\ overline {k}} = (- 1) ^ {k} x ^ {\ underline {k}}}$
${\ displaystyle x ^ {\ underline {k}} = (- x) ^ {\ overline {k}} = 0}$  For ${\ displaystyle 0 \ leq x

Relationships with other known numbers

With the help of the falling factorials, the binomial coefficients can be defined in general:

${\ displaystyle {\ binom {x} {k}}: = {\ frac {1} {k!}} x ^ {\ underline {k}}}$

The following equations also apply, where and denote the (unsigned) Stirling numbers of the first and second kind: ${\ displaystyle \ displaystyle \ left [{n \ atop k} \ right]}$${\ displaystyle \ displaystyle \ left \ {\! {n \ atop k} \! \ right \}}$

${\ displaystyle \ displaystyle x ^ {k} = \ sum _ {j = - \ infty} ^ {\ infty} \ left \ {\! {k \ atop j} \! \ right \} \ cdot x ^ {\ underline {j}}}$
${\ displaystyle \ displaystyle x ^ {k} = \ sum _ {j = - \ infty} ^ {\ infty} (- 1) ^ {kj} \ left \ {{k \ atop j} \ right \} \ cdot x ^ {\ overline {j}}}$
${\ displaystyle \ displaystyle x ^ {\ overline {k}} = \ sum _ {j = - \ infty} ^ {\ infty} \ left [{k \ atop j} \ right] \ cdot x ^ {j}}$
${\ displaystyle \ displaystyle x ^ {\ underline {k}} = \ sum _ {j = - \ infty} ^ {\ infty} (- 1) ^ {kj} \ left [{k \ atop j} \ right] \ cdot x ^ {j}}$

Occurrence in analysis

${\ displaystyle {{\ operatorname {d} ^ {j}} \ over \ operatorname {d} \! x ^ {j}} x ^ {k} = k ^ {\ underline {j}} x ^ {kj} }$