Factorials Examples Taylor Series Expansions Extensions
Generalized Factorials and Taylor Expansions
Michael R. Pilla Winona State University
April 16, 2010
Michael R. Pilla Generalizations of the Factorial Function (n−0)(n−1)(n−2)···(n−(n−2))(n−(n−1)).
For S = {a0, a1, a2, a3, ...} ⊆ N,
Define n!S = (an − a0)(an − a1) ··· (an − an−1).
Cons: Pros: It is not theoretically We have a nice formula. useful. It works on subsets of N. What about the order of the subset?
Factorials Examples Taylor Series Expansions Extensions
Factorials
Recall the definition of a factorial
n! = n·(n−1)···2·1 =
Michael R. Pilla Generalizations of the Factorial Function For S = {a0, a1, a2, a3, ...} ⊆ N,
Define n!S = (an − a0)(an − a1) ··· (an − an−1).
Cons: Pros: It is not theoretically We have a nice formula. useful. It works on subsets of N. What about the order of the subset?
Factorials Examples Taylor Series Expansions Extensions
Factorials
Recall the definition of a factorial
n! = n·(n−1)···2·1 = (n−0)(n−1)(n−2)···(n−(n−2))(n−(n−1)).
Michael R. Pilla Generalizations of the Factorial Function Cons: Pros: It is not theoretically We have a nice formula. useful. It works on subsets of N. What about the order of the subset?
Factorials Examples Taylor Series Expansions Extensions
Factorials
Recall the definition of a factorial
n! = n·(n−1)···2·1 = (n−0)(n−1)(n−2)···(n−(n−2))(n−(n−1)).
For S = {a0, a1, a2, a3, ...} ⊆ N,
Define n!S = (an − a0)(an − a1) ··· (an − an−1).
Michael R. Pilla Generalizations of the Factorial Function Cons: It is not theoretically We have a nice formula. useful. It works on subsets of N. What about the order of the subset?
Factorials Examples Taylor Series Expansions Extensions
Factorials
Recall the definition of a factorial
n! = n·(n−1)···2·1 = (n−0)(n−1)(n−2)···(n−(n−2))(n−(n−1)).
For S = {a0, a1, a2, a3, ...} ⊆ N,
Define n!S = (an − a0)(an − a1) ··· (an − an−1).
Pros:
Michael R. Pilla Generalizations of the Factorial Function Cons: It is not theoretically useful. It works on subsets of N. What about the order of the subset?
Factorials Examples Taylor Series Expansions Extensions
Factorials
Recall the definition of a factorial
n! = n·(n−1)···2·1 = (n−0)(n−1)(n−2)···(n−(n−2))(n−(n−1)).
For S = {a0, a1, a2, a3, ...} ⊆ N,
Define n!S = (an − a0)(an − a1) ··· (an − an−1).
Pros: We have a nice formula.
Michael R. Pilla Generalizations of the Factorial Function Cons: It is not theoretically useful. What about the order of the subset?
Factorials Examples Taylor Series Expansions Extensions
Factorials
Recall the definition of a factorial
n! = n·(n−1)···2·1 = (n−0)(n−1)(n−2)···(n−(n−2))(n−(n−1)).
For S = {a0, a1, a2, a3, ...} ⊆ N,
Define n!S = (an − a0)(an − a1) ··· (an − an−1).
Pros: We have a nice formula. It works on subsets of N.
Michael R. Pilla Generalizations of the Factorial Function It is not theoretically useful. What about the order of the subset?
Factorials Examples Taylor Series Expansions Extensions
Factorials
Recall the definition of a factorial
n! = n·(n−1)···2·1 = (n−0)(n−1)(n−2)···(n−(n−2))(n−(n−1)).
For S = {a0, a1, a2, a3, ...} ⊆ N,
Define n!S = (an − a0)(an − a1) ··· (an − an−1).
Cons: Pros: We have a nice formula. It works on subsets of N.
Michael R. Pilla Generalizations of the Factorial Function What about the order of the subset?
Factorials Examples Taylor Series Expansions Extensions
Factorials
Recall the definition of a factorial
n! = n·(n−1)···2·1 = (n−0)(n−1)(n−2)···(n−(n−2))(n−(n−1)).
For S = {a0, a1, a2, a3, ...} ⊆ N,
Define n!S = (an − a0)(an − a1) ··· (an − an−1).
Cons: Pros: It is not theoretically We have a nice formula. useful. It works on subsets of N.
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Factorials
Recall the definition of a factorial
n! = n·(n−1)···2·1 = (n−0)(n−1)(n−2)···(n−(n−2))(n−(n−1)).
For S = {a0, a1, a2, a3, ...} ⊆ N,
Define n!S = (an − a0)(an − a1) ··· (an − an−1).
Cons: Pros: It is not theoretically We have a nice formula. useful. It works on subsets of N. What about the order of the subset?
Michael R. Pilla Generalizations of the Factorial Function Theorem
If S = {ai } is p-ordered for all primes simultaneously then
n!S = |(an − a0)(an − a1) ··· (an − an−1)|.
If the ordering works for all primes simultaneosly, then we can achieve nice formulas.
If a subset cannot be simultaneously ordered, the formulas are ugly (if and when they exist) and the factorials difficult to calculate. Goal: Extend factorials to “nice”, “natural” subsets of N that have closed formulas.
Factorials Examples Taylor Series Expansions Extensions
Generalized Factorials
Bhargava: Let’s look at prime factorizations and play a game called p-ordering for each prime p.
Michael R. Pilla Generalizations of the Factorial Function Theorem
If S = {ai } is p-ordered for all primes simultaneously then
n!S = |(an − a0)(an − a1) ··· (an − an−1)|.
If a subset cannot be simultaneously ordered, the formulas are ugly (if and when they exist) and the factorials difficult to calculate. Goal: Extend factorials to “nice”, “natural” subsets of N that have closed formulas.
Factorials Examples Taylor Series Expansions Extensions
Generalized Factorials
Bhargava: Let’s look at prime factorizations and play a game called p-ordering for each prime p. If the ordering works for all primes simultaneosly, then we can achieve nice formulas.
Michael R. Pilla Generalizations of the Factorial Function If a subset cannot be simultaneously ordered, the formulas are ugly (if and when they exist) and the factorials difficult to calculate. Goal: Extend factorials to “nice”, “natural” subsets of N that have closed formulas.
Factorials Examples Taylor Series Expansions Extensions
Generalized Factorials
Bhargava: Let’s look at prime factorizations and play a game called p-ordering for each prime p. If the ordering works for all primes simultaneosly, then we can achieve nice formulas. Theorem
If S = {ai } is p-ordered for all primes simultaneously then
n!S = |(an − a0)(an − a1) ··· (an − an−1)|.
Michael R. Pilla Generalizations of the Factorial Function Goal: Extend factorials to “nice”, “natural” subsets of N that have closed formulas.
Factorials Examples Taylor Series Expansions Extensions
Generalized Factorials
Bhargava: Let’s look at prime factorizations and play a game called p-ordering for each prime p. If the ordering works for all primes simultaneosly, then we can achieve nice formulas. Theorem
If S = {ai } is p-ordered for all primes simultaneously then
n!S = |(an − a0)(an − a1) ··· (an − an−1)|.
If a subset cannot be simultaneously ordered, the formulas are ugly (if and when they exist) and the factorials difficult to calculate.
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Generalized Factorials
Bhargava: Let’s look at prime factorizations and play a game called p-ordering for each prime p. If the ordering works for all primes simultaneosly, then we can achieve nice formulas. Theorem
If S = {ai } is p-ordered for all primes simultaneously then
n!S = |(an − a0)(an − a1) ··· (an − an−1)|.
If a subset cannot be simultaneously ordered, the formulas are ugly (if and when they exist) and the factorials difficult to calculate. Goal: Extend factorials to “nice”, “natural” subsets of N that have closed formulas.
Michael R. Pilla Generalizations of the Factorial Function = 711! = 722! = 733!
1!A = (9 − 2) = 7
2!A = (16 − 2)(16 − 9) = 98
3!A = (23 − 2)(23 − 9)(23 − 16) = 2058 n n!A = 7 n!
Example Let S = aN + b of all integers b mod a. The natural ordering is p-ordered for all primes simultaneously. Thus
n n!aN+b = a n!
Factorials Examples Taylor Series Expansions Extensions
Arithmetic Sets Example Consider the arithmetic set A = {2, 9, 16, 23, ...} (i.e. all integers 2 mod 7). This is p-ordered for all primes simultaneously. Thus
Michael R. Pilla Generalizations of the Factorial Function = 722! = 733!
= 711!
2!A = (16 − 2)(16 − 9) = 98
3!A = (23 − 2)(23 − 9)(23 − 16) = 2058 n n!A = 7 n!
Example Let S = aN + b of all integers b mod a. The natural ordering is p-ordered for all primes simultaneously. Thus
n n!aN+b = a n!
Factorials Examples Taylor Series Expansions Extensions
Arithmetic Sets Example Consider the arithmetic set A = {2, 9, 16, 23, ...} (i.e. all integers 2 mod 7). This is p-ordered for all primes simultaneously. Thus
1!A = (9 − 2) = 7
Michael R. Pilla Generalizations of the Factorial Function = 733!
= 711! = 722!
3!A = (23 − 2)(23 − 9)(23 − 16) = 2058 n n!A = 7 n!
Example Let S = aN + b of all integers b mod a. The natural ordering is p-ordered for all primes simultaneously. Thus
n n!aN+b = a n!
Factorials Examples Taylor Series Expansions Extensions
Arithmetic Sets Example Consider the arithmetic set A = {2, 9, 16, 23, ...} (i.e. all integers 2 mod 7). This is p-ordered for all primes simultaneously. Thus
1!A = (9 − 2) = 7
2!A = (16 − 2)(16 − 9) = 98
Michael R. Pilla Generalizations of the Factorial Function = 711! = 722! = 733! n n!A = 7 n!
Example Let S = aN + b of all integers b mod a. The natural ordering is p-ordered for all primes simultaneously. Thus
n n!aN+b = a n!
Factorials Examples Taylor Series Expansions Extensions
Arithmetic Sets Example Consider the arithmetic set A = {2, 9, 16, 23, ...} (i.e. all integers 2 mod 7). This is p-ordered for all primes simultaneously. Thus
1!A = (9 − 2) = 7
2!A = (16 − 2)(16 − 9) = 98
3!A = (23 − 2)(23 − 9)(23 − 16) = 2058
Michael R. Pilla Generalizations of the Factorial Function Example Let S = aN + b of all integers b mod a. The natural ordering is p-ordered for all primes simultaneously. Thus
n n!aN+b = a n!
Factorials Examples Taylor Series Expansions Extensions
Arithmetic Sets Example Consider the arithmetic set A = {2, 9, 16, 23, ...} (i.e. all integers 2 mod 7). This is p-ordered for all primes simultaneously. Thus 1 1!A = (9 − 2) = 7 = 7 1! 2 2!A = (16 − 2)(16 − 9) = 98 = 7 2! 3 3!A = (23 − 2)(23 − 9)(23 − 16) = 2058 = 7 3! n n!A = 7 n!
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Arithmetic Sets Example Consider the arithmetic set A = {2, 9, 16, 23, ...} (i.e. all integers 2 mod 7). This is p-ordered for all primes simultaneously. Thus 1 1!A = (9 − 2) = 7 = 7 1! 2 2!A = (16 − 2)(16 − 9) = 98 = 7 2! 3 3!A = (23 − 2)(23 − 9)(23 − 16) = 2058 = 7 3! n n!A = 7 n!
Example Let S = aN + b of all integers b mod a. The natural ordering is p-ordered for all primes simultaneously. Thus
n n!aN+b = a n!
Michael R. Pilla Generalizations of the Factorial Function 1!Z2 = (1 − 0) = 1
2!Z2 = (4 − 0)(4 − 1) = 12
3!Z2 = (9 − 0)(9 − 1)(9 − 4) = 360
4!Z2 = (16 − 0)(16 − 1)(16 − 4)(16 − 9) = 20160
2 2 2 2 2 n!Z2 = (n − 0)(n − 1)(n − 4) ··· (n − (n − 1) ) = (n − 0)(n + 0)(n − 1)(n + 1)...(n − (n − 1))(n + (n − 1)) 2n = (2n − 1)(2n − 2) ··· (n)(n − 1) ··· (1) 2 (2n)! = 2
Factorials Examples Taylor Series Expansions Extensions
Set of Squares Example 2 The set Z = {0, 1, 4, 9, 16, ...} of square numbers admits a simultaneous p-ordering. Thus
Michael R. Pilla Generalizations of the Factorial Function 2!Z2 = (4 − 0)(4 − 1) = 12
3!Z2 = (9 − 0)(9 − 1)(9 − 4) = 360
4!Z2 = (16 − 0)(16 − 1)(16 − 4)(16 − 9) = 20160
2 2 2 2 2 n!Z2 = (n − 0)(n − 1)(n − 4) ··· (n − (n − 1) ) = (n − 0)(n + 0)(n − 1)(n + 1)...(n − (n − 1))(n + (n − 1)) 2n = (2n − 1)(2n − 2) ··· (n)(n − 1) ··· (1) 2 (2n)! = 2
Factorials Examples Taylor Series Expansions Extensions
Set of Squares Example 2 The set Z = {0, 1, 4, 9, 16, ...} of square numbers admits a simultaneous p-ordering. Thus
1!Z2 = (1 − 0) = 1
Michael R. Pilla Generalizations of the Factorial Function 3!Z2 = (9 − 0)(9 − 1)(9 − 4) = 360
4!Z2 = (16 − 0)(16 − 1)(16 − 4)(16 − 9) = 20160
2 2 2 2 2 n!Z2 = (n − 0)(n − 1)(n − 4) ··· (n − (n − 1) ) = (n − 0)(n + 0)(n − 1)(n + 1)...(n − (n − 1))(n + (n − 1)) 2n = (2n − 1)(2n − 2) ··· (n)(n − 1) ··· (1) 2 (2n)! = 2
Factorials Examples Taylor Series Expansions Extensions
Set of Squares Example 2 The set Z = {0, 1, 4, 9, 16, ...} of square numbers admits a simultaneous p-ordering. Thus
1!Z2 = (1 − 0) = 1
2!Z2 = (4 − 0)(4 − 1) = 12
Michael R. Pilla Generalizations of the Factorial Function 4!Z2 = (16 − 0)(16 − 1)(16 − 4)(16 − 9) = 20160
2 2 2 2 2 n!Z2 = (n − 0)(n − 1)(n − 4) ··· (n − (n − 1) ) = (n − 0)(n + 0)(n − 1)(n + 1)...(n − (n − 1))(n + (n − 1)) 2n = (2n − 1)(2n − 2) ··· (n)(n − 1) ··· (1) 2 (2n)! = 2
Factorials Examples Taylor Series Expansions Extensions
Set of Squares Example 2 The set Z = {0, 1, 4, 9, 16, ...} of square numbers admits a simultaneous p-ordering. Thus
1!Z2 = (1 − 0) = 1
2!Z2 = (4 − 0)(4 − 1) = 12
3!Z2 = (9 − 0)(9 − 1)(9 − 4) = 360
Michael R. Pilla Generalizations of the Factorial Function 2 2 2 2 2 n!Z2 = (n − 0)(n − 1)(n − 4) ··· (n − (n − 1) ) = (n − 0)(n + 0)(n − 1)(n + 1)...(n − (n − 1))(n + (n − 1)) 2n = (2n − 1)(2n − 2) ··· (n)(n − 1) ··· (1) 2 (2n)! = 2
Factorials Examples Taylor Series Expansions Extensions
Set of Squares Example 2 The set Z = {0, 1, 4, 9, 16, ...} of square numbers admits a simultaneous p-ordering. Thus
1!Z2 = (1 − 0) = 1
2!Z2 = (4 − 0)(4 − 1) = 12
3!Z2 = (9 − 0)(9 − 1)(9 − 4) = 360
4!Z2 = (16 − 0)(16 − 1)(16 − 4)(16 − 9) = 20160
Michael R. Pilla Generalizations of the Factorial Function = (n − 0)(n + 0)(n − 1)(n + 1)...(n − (n − 1))(n + (n − 1)) 2n = (2n − 1)(2n − 2) ··· (n)(n − 1) ··· (1) 2 (2n)! = 2
Factorials Examples Taylor Series Expansions Extensions
Set of Squares Example 2 The set Z = {0, 1, 4, 9, 16, ...} of square numbers admits a simultaneous p-ordering. Thus
1!Z2 = (1 − 0) = 1
2!Z2 = (4 − 0)(4 − 1) = 12
3!Z2 = (9 − 0)(9 − 1)(9 − 4) = 360
4!Z2 = (16 − 0)(16 − 1)(16 − 4)(16 − 9) = 20160
2 2 2 2 2 n!Z2 = (n − 0)(n − 1)(n − 4) ··· (n − (n − 1) )
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Set of Squares Example 2 The set Z = {0, 1, 4, 9, 16, ...} of square numbers admits a simultaneous p-ordering. Thus
1!Z2 = (1 − 0) = 1
2!Z2 = (4 − 0)(4 − 1) = 12
3!Z2 = (9 − 0)(9 − 1)(9 − 4) = 360
4!Z2 = (16 − 0)(16 − 1)(16 − 4)(16 − 9) = 20160
2 2 2 2 2 n!Z2 = (n − 0)(n − 1)(n − 4) ··· (n − (n − 1) ) = (n − 0)(n + 0)(n − 1)(n + 1)...(n − (n − 1))(n + (n − 1)) 2n = (2n − 1)(2n − 2) ··· (n)(n − 1) ··· (1) 2 (2n)! = 2
Michael R. Pilla Generalizations of the Factorial Function 1!2T = (2 − 0) = 2
2!2T = (6 − 0)(6 − 2) = 24
3!2T = (12 − 0)(12 − 2)(12 − 6) = 720
4!2T = (20 − 0)(20 − 2)(20 − 6)(20 − 12) = 40320
n!2T = (2n)!
Factorials Examples Taylor Series Expansions Extensions
Twice Triangulars (Squares Modified)
Example Likewise, one can show the set 2 2T = {n + n | n ∈ N} = {0, 2, 6, 12, 20, ...} admits a simultaneous p-ordering. Thus
Michael R. Pilla Generalizations of the Factorial Function 2!2T = (6 − 0)(6 − 2) = 24
3!2T = (12 − 0)(12 − 2)(12 − 6) = 720
4!2T = (20 − 0)(20 − 2)(20 − 6)(20 − 12) = 40320
n!2T = (2n)!
Factorials Examples Taylor Series Expansions Extensions
Twice Triangulars (Squares Modified)
Example Likewise, one can show the set 2 2T = {n + n | n ∈ N} = {0, 2, 6, 12, 20, ...} admits a simultaneous p-ordering. Thus
1!2T = (2 − 0) = 2
Michael R. Pilla Generalizations of the Factorial Function 3!2T = (12 − 0)(12 − 2)(12 − 6) = 720
4!2T = (20 − 0)(20 − 2)(20 − 6)(20 − 12) = 40320
n!2T = (2n)!
Factorials Examples Taylor Series Expansions Extensions
Twice Triangulars (Squares Modified)
Example Likewise, one can show the set 2 2T = {n + n | n ∈ N} = {0, 2, 6, 12, 20, ...} admits a simultaneous p-ordering. Thus
1!2T = (2 − 0) = 2
2!2T = (6 − 0)(6 − 2) = 24
Michael R. Pilla Generalizations of the Factorial Function 4!2T = (20 − 0)(20 − 2)(20 − 6)(20 − 12) = 40320
n!2T = (2n)!
Factorials Examples Taylor Series Expansions Extensions
Twice Triangulars (Squares Modified)
Example Likewise, one can show the set 2 2T = {n + n | n ∈ N} = {0, 2, 6, 12, 20, ...} admits a simultaneous p-ordering. Thus
1!2T = (2 − 0) = 2
2!2T = (6 − 0)(6 − 2) = 24
3!2T = (12 − 0)(12 − 2)(12 − 6) = 720
Michael R. Pilla Generalizations of the Factorial Function n!2T = (2n)!
Factorials Examples Taylor Series Expansions Extensions
Twice Triangulars (Squares Modified)
Example Likewise, one can show the set 2 2T = {n + n | n ∈ N} = {0, 2, 6, 12, 20, ...} admits a simultaneous p-ordering. Thus
1!2T = (2 − 0) = 2
2!2T = (6 − 0)(6 − 2) = 24
3!2T = (12 − 0)(12 − 2)(12 − 6) = 720
4!2T = (20 − 0)(20 − 2)(20 − 6)(20 − 12) = 40320
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Twice Triangulars (Squares Modified)
Example Likewise, one can show the set 2 2T = {n + n | n ∈ N} = {0, 2, 6, 12, 20, ...} admits a simultaneous p-ordering. Thus
1!2T = (2 − 0) = 2
2!2T = (6 − 0)(6 − 2) = 24
3!2T = (12 − 0)(12 − 2)(12 − 6) = 720
4!2T = (20 − 0)(20 − 2)(20 − 6)(20 − 12) = 40320
n!2T = (2n)!
Michael R. Pilla Generalizations of the Factorial Function 1!G = (15 − 5) = 10
2!G = (45 − 5)(45 − 15) = 1200
3!G = (135 − 5)(135 − 15)(135 − 45) = 1404000
4!G = (405−5)(405−15)(405−45)(405−135) = 15163200000
Factorials Examples Taylor Series Expansions Extensions
Example Consider the geometric progression G = {5 · 3n} = {5, 15, 45, 135, 405, ...}. This set admits a simultaneous p-ordering and thus
Michael R. Pilla Generalizations of the Factorial Function 2!G = (45 − 5)(45 − 15) = 1200
3!G = (135 − 5)(135 − 15)(135 − 45) = 1404000
4!G = (405−5)(405−15)(405−45)(405−135) = 15163200000
Factorials Examples Taylor Series Expansions Extensions
A Geometric Progression
Example Consider the geometric progression G = {5 · 3n} = {5, 15, 45, 135, 405, ...}. This set admits a simultaneous p-ordering and thus
1!G = (15 − 5) = 10
Michael R. Pilla Generalizations of the Factorial Function 3!G = (135 − 5)(135 − 15)(135 − 45) = 1404000
4!G = (405−5)(405−15)(405−45)(405−135) = 15163200000
Factorials Examples Taylor Series Expansions Extensions
A Geometric Progression
Example Consider the geometric progression G = {5 · 3n} = {5, 15, 45, 135, 405, ...}. This set admits a simultaneous p-ordering and thus
1!G = (15 − 5) = 10
2!G = (45 − 5)(45 − 15) = 1200
Michael R. Pilla Generalizations of the Factorial Function 4!G = (405−5)(405−15)(405−45)(405−135) = 15163200000
Factorials Examples Taylor Series Expansions Extensions
A Geometric Progression
Example Consider the geometric progression G = {5 · 3n} = {5, 15, 45, 135, 405, ...}. This set admits a simultaneous p-ordering and thus
1!G = (15 − 5) = 10
2!G = (45 − 5)(45 − 15) = 1200
3!G = (135 − 5)(135 − 15)(135 − 45) = 1404000
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
A Geometric Progression
Example Consider the geometric progression G = {5 · 3n} = {5, 15, 45, 135, 405, ...}. This set admits a simultaneous p-ordering and thus
1!G = (15 − 5) = 10
2!G = (45 − 5)(45 − 15) = 1200
3!G = (135 − 5)(135 − 15)(135 − 45) = 1404000
4!G = (405−5)(405−15)(405−45)(405−135) = 15163200000
Michael R. Pilla Generalizations of the Factorial Function n n n n−1 n!aqN = (aq − a)(aq − aq) ··· (aq − aq ) = anqn(1 − q−n)(1 − q1−n) ··· (1 − q−2)(1 − q−1))
n −n(n+1) 2 n = (−aq) q 2 (1 − q)(1 − q ) ··· (1 − q )
n −n(n+1) = (−aq) q 2 (q : q)n
where (q : q)n is the q-Pochhammer symbol.
Factorials Examples Taylor Series Expansions Extensions
A General Geometric Progression
Example The geometric progression aqN gives a simultaneous p-ordering. Thus
Michael R. Pilla Generalizations of the Factorial Function = anqn(1 − q−n)(1 − q1−n) ··· (1 − q−2)(1 − q−1))
n −n(n+1) 2 n = (−aq) q 2 (1 − q)(1 − q ) ··· (1 − q )
n −n(n+1) = (−aq) q 2 (q : q)n
where (q : q)n is the q-Pochhammer symbol.
Factorials Examples Taylor Series Expansions Extensions
A General Geometric Progression
Example The geometric progression aqN gives a simultaneous p-ordering. Thus
n n n n−1 n!aqN = (aq − a)(aq − aq) ··· (aq − aq )
Michael R. Pilla Generalizations of the Factorial Function n −n(n+1) 2 n = (−aq) q 2 (1 − q)(1 − q ) ··· (1 − q )
n −n(n+1) = (−aq) q 2 (q : q)n
where (q : q)n is the q-Pochhammer symbol.
Factorials Examples Taylor Series Expansions Extensions
A General Geometric Progression
Example The geometric progression aqN gives a simultaneous p-ordering. Thus
n n n n−1 n!aqN = (aq − a)(aq − aq) ··· (aq − aq ) = anqn(1 − q−n)(1 − q1−n) ··· (1 − q−2)(1 − q−1))
Michael R. Pilla Generalizations of the Factorial Function n −n(n+1) = (−aq) q 2 (q : q)n
where (q : q)n is the q-Pochhammer symbol.
Factorials Examples Taylor Series Expansions Extensions
A General Geometric Progression
Example The geometric progression aqN gives a simultaneous p-ordering. Thus
n n n n−1 n!aqN = (aq − a)(aq − aq) ··· (aq − aq ) = anqn(1 − q−n)(1 − q1−n) ··· (1 − q−2)(1 − q−1))
n −n(n+1) 2 n = (−aq) q 2 (1 − q)(1 − q ) ··· (1 − q )
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
A General Geometric Progression
Example The geometric progression aqN gives a simultaneous p-ordering. Thus
n n n n−1 n!aqN = (aq − a)(aq − aq) ··· (aq − aq ) = anqn(1 − q−n)(1 − q1−n) ··· (1 − q−2)(1 − q−1))
n −n(n+1) 2 n = (−aq) q 2 (1 − q)(1 − q ) ··· (1 − q )
n −n(n+1) = (−aq) q 2 (q : q)n
where (q : q)n is the q-Pochhammer symbol.
Michael R. Pilla Generalizations of the Factorial Function 3 The set of cubes N does not admit a simultaneous p-ordering and has no nice formula.
{n!N3 } = {1, 2, 504, 504, 35280, ...}
b n−1 c+b n−1 c+b n−1 c+··· Y p−1 p(p−1) p2(p−1) n!P = p p
{n!P} = {1, 2, 24, 48, 5670, ...} Example
Factorials Examples Taylor Series Expansions Extensions
Non-Simultaneous Sets
Example While it can be shown that the primes do NOT admit a simultaneous p-ordering, Bhargava gives an intricate formula to calculate them.
Michael R. Pilla Generalizations of the Factorial Function 3 The set of cubes N does not admit a simultaneous p-ordering and has no nice formula.
{n!N3 } = {1, 2, 504, 504, 35280, ...}
{n!P} = {1, 2, 24, 48, 5670, ...} Example
Factorials Examples Taylor Series Expansions Extensions
Non-Simultaneous Sets
Example While it can be shown that the primes do NOT admit a simultaneous p-ordering, Bhargava gives an intricate formula to calculate them.
b n−1 c+b n−1 c+b n−1 c+··· Y p−1 p(p−1) p2(p−1) n!P = p p
Michael R. Pilla Generalizations of the Factorial Function 3 The set of cubes N does not admit a simultaneous p-ordering and has no nice formula.
{n!N3 } = {1, 2, 504, 504, 35280, ...}
Example
Factorials Examples Taylor Series Expansions Extensions
Non-Simultaneous Sets
Example While it can be shown that the primes do NOT admit a simultaneous p-ordering, Bhargava gives an intricate formula to calculate them.
b n−1 c+b n−1 c+b n−1 c+··· Y p−1 p(p−1) p2(p−1) n!P = p p
{n!P} = {1, 2, 24, 48, 5670, ...}
Michael R. Pilla Generalizations of the Factorial Function {n!N3 } = {1, 2, 504, 504, 35280, ...}
Factorials Examples Taylor Series Expansions Extensions
Non-Simultaneous Sets
Example While it can be shown that the primes do NOT admit a simultaneous p-ordering, Bhargava gives an intricate formula to calculate them.
b n−1 c+b n−1 c+b n−1 c+··· Y p−1 p(p−1) p2(p−1) n!P = p p
{n!P} = {1, 2, 24, 48, 5670, ...} Example 3 The set of cubes N does not admit a simultaneous p-ordering and has no nice formula.
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Non-Simultaneous Sets
Example While it can be shown that the primes do NOT admit a simultaneous p-ordering, Bhargava gives an intricate formula to calculate them.
b n−1 c+b n−1 c+b n−1 c+··· Y p−1 p(p−1) p2(p−1) n!P = p p
{n!P} = {1, 2, 24, 48, 5670, ...} Example 3 The set of cubes N does not admit a simultaneous p-ordering and has no nice formula.
{n!N3 } = {1, 2, 504, 504, 35280, ...}
Michael R. Pilla Generalizations of the Factorial Function n aN + b ! n!aN+b = a n!
2T ! n!2T = (2n)!
2 (2n)! Z ! n!Z2 = 2
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n Q (stuff ) P ! n!P = p p
Factorials Examples Taylor Series Expansions Extensions
Summary of Examples
N ! n!N = n!
Michael R. Pilla Generalizations of the Factorial Function 2T ! n!2T = (2n)!
2 (2n)! Z ! n!Z2 = 2
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n Q (stuff ) P ! n!P = p p
Factorials Examples Taylor Series Expansions Extensions
Summary of Examples
N ! n!N = n! n aN + b ! n!aN+b = a n!
Michael R. Pilla Generalizations of the Factorial Function 2 (2n)! Z ! n!Z2 = 2
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n Q (stuff ) P ! n!P = p p
Factorials Examples Taylor Series Expansions Extensions
Summary of Examples
N ! n!N = n! n aN + b ! n!aN+b = a n!
2T ! n!2T = (2n)!
Michael R. Pilla Generalizations of the Factorial Function −n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n Q (stuff ) P ! n!P = p p
Factorials Examples Taylor Series Expansions Extensions
Summary of Examples
N ! n!N = n! n aN + b ! n!aN+b = a n!
2T ! n!2T = (2n)!
2 (2n)! Z ! n!Z2 = 2
Michael R. Pilla Generalizations of the Factorial Function Q (stuff ) P ! n!P = p p
Factorials Examples Taylor Series Expansions Extensions
Summary of Examples
N ! n!N = n! n aN + b ! n!aN+b = a n!
2T ! n!2T = (2n)!
2 (2n)! Z ! n!Z2 = 2
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Summary of Examples
N ! n!N = n! n aN + b ! n!aN+b = a n!
2T ! n!2T = (2n)!
2 (2n)! Z ! n!Z2 = 2
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n Q (stuff ) P ! n!P = p p
Michael R. Pilla Generalizations of the Factorial Function Recall that one can write ex as a Taylor series. That is
∞ X xn ex = n! n=0
Question: What is the !S -analogue to this equation?
As it turns out, this is the wrong question to ask.
Factorials Examples Taylor Series Expansions Extensions
Question
Michael R. Pilla Generalizations of the Factorial Function That is
∞ X xn ex = n! n=0
Question: What is the !S -analogue to this equation?
As it turns out, this is the wrong question to ask.
Factorials Examples Taylor Series Expansions Extensions
Question
Recall that one can write ex as a Taylor series.
Michael R. Pilla Generalizations of the Factorial Function Question: What is the !S -analogue to this equation?
As it turns out, this is the wrong question to ask.
Factorials Examples Taylor Series Expansions Extensions
Question
Recall that one can write ex as a Taylor series. That is
∞ X xn ex = n! n=0
Michael R. Pilla Generalizations of the Factorial Function As it turns out, this is the wrong question to ask.
Factorials Examples Taylor Series Expansions Extensions
Question
Recall that one can write ex as a Taylor series. That is
∞ X xn ex = n! n=0
Question: What is the !S -analogue to this equation?
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Question
Recall that one can write ex as a Taylor series. That is
∞ X xn ex = n! n=0
Question: What is the !S -analogue to this equation?
As it turns out, this is the wrong question to ask.
Michael R. Pilla Generalizations of the Factorial Function ∞ X mn m m2 m3 (ex )m = emx = xn = 1 + x + x2 + x3 + ··· n! 1! 2! 3! n=0
Right Question: What is the !S -analogue of this equation?
Goal: 1) The numerator of each “coefficient” is a polynomial in m.
2) The denominator of each “coefficient” is a factorial.
Factorials Examples Taylor Series Expansions Extensions
Right Question
Raising to the power of m, we can write (ex )m as follows:
Michael R. Pilla Generalizations of the Factorial Function Right Question: What is the !S -analogue of this equation?
Goal: 1) The numerator of each “coefficient” is a polynomial in m.
2) The denominator of each “coefficient” is a factorial.
Factorials Examples Taylor Series Expansions Extensions
Right Question
Raising to the power of m, we can write (ex )m as follows:
∞ X mn m m2 m3 (ex )m = emx = xn = 1 + x + x2 + x3 + ··· n! 1! 2! 3! n=0
Michael R. Pilla Generalizations of the Factorial Function Goal: 1) The numerator of each “coefficient” is a polynomial in m.
2) The denominator of each “coefficient” is a factorial.
Factorials Examples Taylor Series Expansions Extensions
Right Question
Raising to the power of m, we can write (ex )m as follows:
∞ X mn m m2 m3 (ex )m = emx = xn = 1 + x + x2 + x3 + ··· n! 1! 2! 3! n=0
Right Question: What is the !S -analogue of this equation?
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Right Question
Raising to the power of m, we can write (ex )m as follows:
∞ X mn m m2 m3 (ex )m = emx = xn = 1 + x + x2 + x3 + ··· n! 1! 2! 3! n=0
Right Question: What is the !S -analogue of this equation?
Goal: 1) The numerator of each “coefficient” is a polynomial in m.
2) The denominator of each “coefficient” is a factorial.
Michael R. Pilla Generalizations of the Factorial Function m m(m−1) 2 m(m−1)(m−2) 3 = 1 + a x + 2a2 x + 6a3 x
m(m−1)(m−2)(m−3) 4 m(m−1)(m−2)(m−3)(m−4) 5 + 24a4 x + 120a5 x + ···
P∞ PaN+b,n(m) n = n=0 x n!aN+b
Notice our numerators are polynomials in m and our denominators n are a n! = n!aN+b.
Factorials Examples Taylor Series Expansions Extensions
(aN + b) -analogue
Example a m x −m ( a−x ) = (1 − a )
Michael R. Pilla Generalizations of the Factorial Function P∞ PaN+b,n(m) n = n=0 x n!aN+b
Notice our numerators are polynomials in m and our denominators n are a n! = n!aN+b.
Factorials Examples Taylor Series Expansions Extensions
(aN + b) -analogue
Example a m x −m ( a−x ) = (1 − a )
m m(m−1) 2 m(m−1)(m−2) 3 = 1 + a x + 2a2 x + 6a3 x
m(m−1)(m−2)(m−3) 4 m(m−1)(m−2)(m−3)(m−4) 5 + 24a4 x + 120a5 x + ···
Michael R. Pilla Generalizations of the Factorial Function P∞ PaN+b,n(m) n = n=0 x n!aN+b
Factorials Examples Taylor Series Expansions Extensions
(aN + b) -analogue
Example a m x −m ( a−x ) = (1 − a )
m m(m−1) 2 m(m−1)(m−2) 3 = 1 + a x + 2a2 x + 6a3 x
m(m−1)(m−2)(m−3) 4 m(m−1)(m−2)(m−3)(m−4) 5 + 24a4 x + 120a5 x + ···
Notice our numerators are polynomials in m and our denominators n are a n! = n!aN+b.
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
(aN + b) -analogue
Example a m x −m ( a−x ) = (1 − a )
m m(m−1) 2 m(m−1)(m−2) 3 = 1 + a x + 2a2 x + 6a3 x
m(m−1)(m−2)(m−3) 4 m(m−1)(m−2)(m−3)(m−4) 5 + 24a4 x + 120a5 x + ···
P∞ PaN+b,n(m) n = n=0 x n!aN+b
Notice our numerators are polynomials in m and our denominators n are a n! = n!aN+b.
Michael R. Pilla Generalizations of the Factorial Function m m+3m(m−1) 2 1 − 2 x + 24 x
15m(m−1)+m+15m(m−1)(m−2) 3 − 720 x + ···
P∞ P2T,n(m) n = n=0 x n!2T 2 Note that by allowing multiplication by scalars, the Z -analogue is ∞ √ X P 2,n(m) 2cosm( x) = Z xn. n! 2 n=0 Z
Factorials Examples Taylor Series Expansions Extensions
2T -analogue
Example √ cosm( x) =
Michael R. Pilla Generalizations of the Factorial Function P∞ P2T,n(m) n = n=0 x n!2T 2 Note that by allowing multiplication by scalars, the Z -analogue is ∞ √ X P 2,n(m) 2cosm( x) = Z xn. n! 2 n=0 Z
Factorials Examples Taylor Series Expansions Extensions
2T -analogue
Example m √ m m+3m(m−1) 2 cos ( x) = 1 − 2 x + 24 x
15m(m−1)+m+15m(m−1)(m−2) 3 − 720 x + ···
Michael R. Pilla Generalizations of the Factorial Function 2 Note that by allowing multiplication by scalars, the Z -analogue is ∞ √ X P 2,n(m) 2cosm( x) = Z xn. n! 2 n=0 Z
Factorials Examples Taylor Series Expansions Extensions
2T -analogue
Example m √ m m+3m(m−1) 2 cos ( x) = 1 − 2 x + 24 x
15m(m−1)+m+15m(m−1)(m−2) 3 − 720 x + ···
P∞ P2T,n(m) n = n=0 x n!2T
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
2T -analogue
Example m √ m m+3m(m−1) 2 cos ( x) = 1 − 2 x + 24 x
15m(m−1)+m+15m(m−1)(m−2) 3 − 720 x + ···
P∞ P2T,n(m) n = n=0 x n!2T 2 Note that by allowing multiplication by scalars, the Z -analogue is ∞ √ X P 2,n(m) 2cosm( x) = Z xn. n! 2 n=0 Z
Michael R. Pilla Generalizations of the Factorial Function ∞ X P ,n(m) = P xn n! n=0 P
Notice that the coefficients in the denominator seem to be the same as the factorials for the set of primes. It turns out that this is so (Chabert, 2005).
Factorials Examples Taylor Series Expansions Extensions
P -analogue
Example
m ln(1 − x) 2 − = 1 + m x + m(3m+5) x2 + m(m +5m+6) x3 + ··· x 2 24 48
Michael R. Pilla Generalizations of the Factorial Function ∞ X P ,n(m) = P xn n! n=0 P
It turns out that this is so (Chabert, 2005).
Factorials Examples Taylor Series Expansions Extensions
P -analogue
Example
m ln(1 − x) 2 − = 1 + m x + m(3m+5) x2 + m(m +5m+6) x3 + ··· x 2 24 48
Notice that the coefficients in the denominator seem to be the same as the factorials for the set of primes.
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
P -analogue
Example
m ln(1 − x) 2 − = 1 + m x + m(3m+5) x2 + m(m +5m+6) x3 + ··· x 2 24 48 ∞ X P ,n(m) = P xn n! n=0 P
Notice that the coefficients in the denominator seem to be the same as the factorials for the set of primes. It turns out that this is so (Chabert, 2005).
Michael R. Pilla Generalizations of the Factorial Function (ex )m
a m ( a−x ) √ cosm( x) √ 2cosm( x)
? m ln(1−x) − x
−1 m ? ! n!? ! (tan (x))
Factorials Examples Taylor Series Expansions Extensions
Summary of !S -analogues
N ! n!N = n! ! n aN + b ! n!aN+b = a n! !
2T ! n!2T = (2n)! !
2 (2n)! Z ! n!Z2 = 2 !
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n !
Q (stuff ) P ! n!P = p p !
Michael R. Pilla Generalizations of the Factorial Function a m ( a−x ) √ cosm( x) √ 2cosm( x)
? m ln(1−x) − x
−1 m ? ! n!? ! (tan (x))
Factorials Examples Taylor Series Expansions Extensions
Summary of !S -analogues
x m N ! n!N = n! ! (e ) n aN + b ! n!aN+b = a n! !
2T ! n!2T = (2n)! !
2 (2n)! Z ! n!Z2 = 2 !
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n !
Q (stuff ) P ! n!P = p p !
Michael R. Pilla Generalizations of the Factorial Function √ cosm( x) √ 2cosm( x)
? m ln(1−x) − x
−1 m ? ! n!? ! (tan (x))
Factorials Examples Taylor Series Expansions Extensions
Summary of !S -analogues
x m N ! n!N = n! ! (e ) n a m aN + b ! n!aN+b = a n! ! ( a−x )
2T ! n!2T = (2n)! !
2 (2n)! Z ! n!Z2 = 2 !
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n !
Q (stuff ) P ! n!P = p p !
Michael R. Pilla Generalizations of the Factorial Function √ 2cosm( x)
? m ln(1−x) − x
−1 m ? ! n!? ! (tan (x))
Factorials Examples Taylor Series Expansions Extensions
Summary of !S -analogues
x m N ! n!N = n! ! (e ) n a m aN + b ! n!aN+b = a n! ! ( a−x ) m √ 2T ! n!2T = (2n)! ! cos ( x)
2 (2n)! Z ! n!Z2 = 2 !
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n !
Q (stuff ) P ! n!P = p p !
Michael R. Pilla Generalizations of the Factorial Function ? m ln(1−x) − x
−1 m ? ! n!? ! (tan (x))
Factorials Examples Taylor Series Expansions Extensions
Summary of !S -analogues
x m N ! n!N = n! ! (e ) n a m aN + b ! n!aN+b = a n! ! ( a−x ) m √ 2T ! n!2T = (2n)! ! cos ( x)
2 (2n)! m √ Z ! n!Z2 = 2 ! 2cos ( x)
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n !
Q (stuff ) P ! n!P = p p !
Michael R. Pilla Generalizations of the Factorial Function ?
−1 m ? ! n!? ! (tan (x))
Factorials Examples Taylor Series Expansions Extensions
Summary of !S -analogues
x m N ! n!N = n! ! (e ) n a m aN + b ! n!aN+b = a n! ! ( a−x ) m √ 2T ! n!2T = (2n)! ! cos ( x)
2 (2n)! m √ Z ! n!Z2 = 2 ! 2cos ( x)
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n ! m Q (stuff ) ln(1−x) P ! n!P = p p ! − x
Michael R. Pilla Generalizations of the Factorial Function −1 m ? ! n!? ! (tan (x))
Factorials Examples Taylor Series Expansions Extensions
Summary of !S -analogues
x m N ! n!N = n! ! (e ) n a m aN + b ! n!aN+b = a n! ! ( a−x ) m √ 2T ! n!2T = (2n)! ! cos ( x)
2 (2n)! m √ Z ! n!Z2 = 2 ! 2cos ( x)
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n ! ? m Q (stuff ) ln(1−x) P ! n!P = p p ! − x
Michael R. Pilla Generalizations of the Factorial Function ? ! n!? !
Factorials Examples Taylor Series Expansions Extensions
Summary of !S -analogues
x m N ! n!N = n! ! (e ) n a m aN + b ! n!aN+b = a n! ! ( a−x ) m √ 2T ! n!2T = (2n)! ! cos ( x)
2 (2n)! m √ Z ! n!Z2 = 2 ! 2cos ( x)
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n ! ? m Q (stuff ) ln(1−x) P ! n!P = p p ! − x
(tan−1(x))m
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Summary of !S -analogues
x m N ! n!N = n! ! (e ) n a m aN + b ! n!aN+b = a n! ! ( a−x ) m √ 2T ! n!2T = (2n)! ! cos ( x)
2 (2n)! m √ Z ! n!Z2 = 2 ! 2cos ( x)
−n(n+1) N n 2 aq ! n!aqN = (−aq) q (q : q)n ! ? m Q (stuff ) ln(1−x) P ! n!P = p p ! − x
−1 m ? ! n!? ! (tan (x))
Michael R. Pilla Generalizations of the Factorial Function (with conditions?)
Every subset of N corresponds to a function. Issues: They probably won’t be classical functions. Difficulty: Finding the correct polynomials in the numerator. 2 Is it theoretically useful to allow for constants such as for Z ?
Every analytic function corresponds to a subset of N. Issues: What are the conditions?
Factorials Examples Taylor Series Expansions Extensions
Future Work
Conjecture A
Conjecture B
Michael R. Pilla Generalizations of the Factorial Function Issues: They probably won’t be classical functions. Difficulty: Finding the correct polynomials in the numerator. 2 Is it theoretically useful to allow for constants such as for Z ?
(with conditions?)
Issues: What are the conditions?
Factorials Examples Taylor Series Expansions Extensions
Future Work
Conjecture A Every subset of N corresponds to a function.
Conjecture B Every analytic function corresponds to a subset of N.
Michael R. Pilla Generalizations of the Factorial Function They probably won’t be classical functions. Difficulty: Finding the correct polynomials in the numerator. 2 Is it theoretically useful to allow for constants such as for Z ?
(with conditions?)
Issues: What are the conditions?
Factorials Examples Taylor Series Expansions Extensions
Future Work
Conjecture A Every subset of N corresponds to a function. Issues:
Conjecture B Every analytic function corresponds to a subset of N.
Michael R. Pilla Generalizations of the Factorial Function Difficulty: Finding the correct polynomials in the numerator. 2 Is it theoretically useful to allow for constants such as for Z ?
(with conditions?)
Issues: What are the conditions?
Factorials Examples Taylor Series Expansions Extensions
Future Work
Conjecture A Every subset of N corresponds to a function. Issues: They probably won’t be classical functions.
Conjecture B Every analytic function corresponds to a subset of N.
Michael R. Pilla Generalizations of the Factorial Function 2 Is it theoretically useful to allow for constants such as for Z ?
(with conditions?)
Issues: What are the conditions?
Factorials Examples Taylor Series Expansions Extensions
Future Work
Conjecture A Every subset of N corresponds to a function. Issues: They probably won’t be classical functions. Difficulty: Finding the correct polynomials in the numerator.
Conjecture B Every analytic function corresponds to a subset of N.
Michael R. Pilla Generalizations of the Factorial Function (with conditions?)
Issues: What are the conditions?
Factorials Examples Taylor Series Expansions Extensions
Future Work
Conjecture A Every subset of N corresponds to a function. Issues: They probably won’t be classical functions. Difficulty: Finding the correct polynomials in the numerator. 2 Is it theoretically useful to allow for constants such as for Z ?
Conjecture B Every analytic function corresponds to a subset of N.
Michael R. Pilla Generalizations of the Factorial Function (with conditions?)
What are the conditions?
Factorials Examples Taylor Series Expansions Extensions
Future Work
Conjecture A Every subset of N corresponds to a function. Issues: They probably won’t be classical functions. Difficulty: Finding the correct polynomials in the numerator. 2 Is it theoretically useful to allow for constants such as for Z ?
Conjecture B Every analytic function corresponds to a subset of N. Issues:
Michael R. Pilla Generalizations of the Factorial Function What are the conditions?
Factorials Examples Taylor Series Expansions Extensions
Future Work
Conjecture A Every subset of N corresponds to a function. Issues: They probably won’t be classical functions. Difficulty: Finding the correct polynomials in the numerator. 2 Is it theoretically useful to allow for constants such as for Z ?
Conjecture B Every analytic function (with conditions?) corresponds to a subset of N. Issues:
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Future Work
Conjecture A Every subset of N corresponds to a function. Issues: They probably won’t be classical functions. Difficulty: Finding the correct polynomials in the numerator. 2 Is it theoretically useful to allow for constants such as for Z ?
Conjecture B Every analytic function (with conditions?) corresponds to a subset of N. Issues: What are the conditions?
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
If The Conjectures Are True...
−a ln(a − x) / ? O R dx a o / a + b a − x N
d dx a (a − x)2 / ?
Michael R. Pilla Generalizations of the Factorial Function Thanks Professor Eric Errthum!
Questions
Factorials Examples Taylor Series Expansions Extensions
Thanks
References
Bhargava, M. (2000). The factorial function and generalizations. The American Mathematical Monthly, 107(9), 783-799.
Chabert, J.L. (2007). Integer-valued polynomials on prime numbers and logarithm power expansion. European Journal of Combinatorics, 28(3), 754-761.
Michael R. Pilla Generalizations of the Factorial Function Questions
Factorials Examples Taylor Series Expansions Extensions
Thanks
References
Bhargava, M. (2000). The factorial function and generalizations. The American Mathematical Monthly, 107(9), 783-799.
Chabert, J.L. (2007). Integer-valued polynomials on prime numbers and logarithm power expansion. European Journal of Combinatorics, 28(3), 754-761.
Thanks Professor Eric Errthum!
Michael R. Pilla Generalizations of the Factorial Function Factorials Examples Taylor Series Expansions Extensions
Thanks
References
Bhargava, M. (2000). The factorial function and generalizations. The American Mathematical Monthly, 107(9), 783-799.
Chabert, J.L. (2007). Integer-valued polynomials on prime numbers and logarithm power expansion. European Journal of Combinatorics, 28(3), 754-761.
Thanks Professor Eric Errthum!
Questions
Michael R. Pilla Generalizations of the Factorial Function