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 = fa0; a1; a2; a3; :::g ⊆ 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 = fa0; a1; a2; a3; :::g ⊆ 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 = fa0; a1; a2; a3; :::g ⊆ 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 = fa0; a1; a2; a3; :::g ⊆ 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 = fa0; a1; a2; a3; :::g ⊆ 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 = fa0; a1; a2; a3; :::g ⊆ 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 = fa0; a1; a2; a3; :::g ⊆ 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 = fa0; a1; a2; a3; :::g ⊆ 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 = fa0; a1; a2; a3; :::g ⊆ 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 = fai g is p-ordered for all primes simultaneously then n!S = j(an − a0)(an − a1) ··· (an − an−1)j: 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 = fai g is p-ordered for all primes simultaneously then n!S = j(an − a0)(an − a1) ··· (an − an−1)j: 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 = fai g is p-ordered for all primes simultaneously then n!S = j(an − a0)(an − a1) ··· (an − an−1)j: 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 = fai g is p-ordered for all primes simultaneously then n!S = j(an − a0)(an − a1) ··· (an − an−1)j: 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 = fai g is p-ordered for all primes simultaneously then n!S = j(an − a0)(an − a1) ··· (an − an−1)j: 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 = f2; 9; 16; 23; :::g (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 = f2; 9; 16; 23; :::g (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 = f2; 9; 16; 23; :::g (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.