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The Generalized Superfactorial, Hyperfactorial and Primorial Functions THE GENERALIZED SUPERFACTORIAL, HYPERFACTORIAL AND PRIMORIAL FUNCTIONS VIGNESH RAMAN Abstract. This paper introduces a new generalized superfactorial function (referable to as nth- degree superfactorial: sf (n)(x)) and a generalized hyper- factorial function (referable to as nth- degree hyperfactorial: H(n)(x)), and we show that these functions possess explicit formulae involving figurate numbers. Besides discussing additional number patterns, we also introduce a generalized primorial function and 2 related theorems. Note that the superfactorial def- inition offered by Sloane and Plouffe (1995) is the definition considered (and not Clifford Pickover's (1995) superfactorial function: n$). 1. Introduction Factorials, their numerous extensions into the complex region 1 and related func- tions 2 find themselves numerous applications in Combinatorics, Number Theory (NT), Functional Analysis, Probability Theory - the list goes on. Unfortunately, not much research has been done on factorial-related functions in recent literature due to two possible reasons: (1) Current research in NT is more oriented towards Analytical NT. (2) Current requirements for research and advancement in Combinatorics, NT and Diophantine Analysis (including algebraic manipulations) are satisfied by the current knowledge possessed about factorials and its closely related functions; it may be so that effort of the mathematical community is being solely directed towards research about functions behaving as extensions of the factorial function into other number systems (eg: Luschny factorial), owing to their ready utility in applied math/real world applications. Research pertaining to r-simplex (figurate) numbers and factorial-related functions arXiv:2012.00882v1 [math.NT] 1 Dec 2020 motivated the investigation into their possible relation to extensions of the super- factorial and hyperfactorial functions (formulated in this very paper). We review certain factorial-related functions and figurate numbers in section 2, followed by the introduction of the generalized superfactorial function and a proof for its explicit formula in section 3. Section 4 will feature a similar treatment of the generalized hyperfactorial function. Section 5 will visually elucidate the aforementioned functions and explore two other number product-patterns involving factorials, and Section 6 will focus on 1Hadamard's Gamma function: Γ(z) and the Luschny Factorial function: L(z) 2Superfactorials: sf(n); Hyperfactorials: H(n); Multifactorials: n!(k) 1 2 VIGNESH RAMAN an extension of the primorial function. Section 7 involves application of group theory and modular arithmetic lemmas to formulate two theorems concerning the superfactorial functions. Finally, Section 8 introduces an extension of the Legendre formula to shed light on the factorization of generalized superfactorials. 2. Factorial-related functions and Figurate Numbers 2.1. The Superfactorial Function. Definition 2.1. The superfactorial function [1] is defined as the product of the first n factorials, i.e. n Y + (2.1) sf(n) = k! n 2 Z ; sf(0) = 1 k=1 It can also be defined recursively as follows: + sf(n + 1) = (n + 1)!(sf(n)); n + 1 2 Z ; sf(0) = 1 Supplementaries: • It is analogous to the tetrahedral numbers,- it represents the products of factorials akin to the tetrahedral numbers representing the sum of summa- tions of successive natural numbers. • It can also alternatively be represented as (on refactorization): n Y n−k+1 + sf(n) = k ; n 2 Z k=1 • Its asymptotic growth is approximately[2]: ζ0(−1)− 3 − 3 − 3 1 + 1 1 +n+ 5 e 4 4n2 2n ∗ (2π) 2 2n (n + 1) 2n2 12 2.2. The Hyperfactorial function. Definition 2.2. The hyperfactorial function[3] is defined as the product of numbers from 1 to n raised to the power of themselves, i.e n Y k + (2.2) H(n) = k ; n 2 Z k=1 It can also be defined as follows: n Y n! H(n) = nP ; n 2 +; nP = k Z k (n − k)! k=1 Supplementaries: • The hyperfactorial sequence for natural numbers gives the discriminants for the probabilists' Hermite polynomial[4]: 2 tx− tx n! I e 2 Hen (x) = n+1 2πi C t • Its asymptotic growth rate is approximately[5]: 2 2 An(6n +6n+1)=12e−n =4; A ! Glaisher Kinkelin constant THE GENERALIZED SUPERFACTORIAL, HYPERFACTORIAL AND PRIMORIAL FUNCTIONS3 Figure 1. Polygonal numbers 2.3. Figurate numbers. Definition 2.3. Figurate numbers are those numbers that can be represented by a regular geometrical arrangement of equally spaced points. Supplementaries: • The nth regular r-polytopic number (i.e. figurate numbers for the r-dimensional analogs of triangles) is given by[6]: n + r − 1 (2.3) P (n) = r r • The above formula essentially implies that (can be verified on addition of each term's respective binomial-coefficient representation): k X (2.4) Pr+1(k) = Pr(i) i=1 • If the arrangement forms a regular polygon, the number is called a polyg- onal number. (Figure 1)[7]. 3. The Generalized Superfactorial Function - GSF Definition 3.1. Define the nth-degree superfactorial as follows: x (n) Y (n−1) + (0) (3.1) sf (x) = sf (k); n 2 Z ; sf (x) ≡ x! k=1 We see that the first degree superfactorial is equivalent to the superfactorial defi- nition given by Sloane and Plouffe (1995): x (1) Y + sf (x) = sf(x) = k!; x 2 Z k=1 We now give an explicit formula for the GSF. Theorem 3.2. The generalized superfactorial function is related to the r-simplex numbers (i.e.figurate numbers, Pn(k)) as follows: x x (n) Y P (k) Y P ((x−k)+1) + ≥ sf (x) = [(x − k) + 1] n = k n ; x 2 Z ; n 2 Z k=1 k=1 4 VIGNESH RAMAN Proof. Theorem 3.2. is proven by using the Principle of Mathematical Induction. Base Case: n = 0 x Y sf (0)(x) = x! = k1 [As noted in section 2.1] k=1 x Y P0((x−k)+1) = k [From equation (2.3), P0(n) = 1] k=1 = 1121 ··· (x − 1)1x1 = x1(x − 1)1 ··· 2111 = xP0(1)(x − 1)P0(2)(x − 2)P0(3) ··· 2P0(x−1)1P0(x) x Y = [(x − k) + 1]P0(k) k=1 Inductive Hypothesis: x Y sf (m)(x) = [(x − k) + 1]Pm(k) k=1 To prove: x x Y Y sf (m+1)(x) = [(x − k) + 1]Pm+1(k) = kPm+1((x−k)+1) k=1 k=1 Now, x Y sf (m+1)(x) = sf (m)(k) [From equation (3.1)] k=1 x " k # Y Y = [(k − i) + 1]Pm(i) [From Inductive Hypothesis] k=1 i=1 1 2 x Y Y Y = [(1 − i) + 1]Pm(i) [(2 − i) + 1]Pm(i) ··· [(x − i) + 1]Pm(i) i=1 i=1 i=1 h i h i h i h i = 1Pm(1) 2Pm(1)1Pm(2) 3Pm(1)2Pm(2)1Pm(3) ··· xPm(1)(x − 1)Pm(2)(x − 2)Pm(3) ··· 1Pm(x) 2 x 3 2x−1 3 2 1 3 X X X 6 7 6 7 6 7 4 Pm(k)5 4 Pm(k)5 4 Pm(k)5 = 1 k=1 2 k=1 ··· x k=1 [On rearranging terms] h i h i = 1Pm+1(x) 2Pm+1(x−1) ··· xPm+1(1) [From equation 2.4] x x Y Y = kPm+1((x−k)+1) = [(x − k) + 1]Pm+1(k) k=1 k=1 THE GENERALIZED SUPERFACTORIAL, HYPERFACTORIAL AND PRIMORIAL FUNCTIONS5 4. The Generalized Hyperfactorial Function - GHF Definition 4.1. Define the nth-degree hyperfactorial as follows: x x Y Y (4.1) H(n)(x) = H(n−1)(k); H(1)(x) ≡ H(x) = kk k=1 k=1 We now give an explicit formula for the GHF. Theorem 4.2. The generalized hyperfactorial function is related to the r-simplex numbers as follows: x (n) Y k[P ((x−k)+1)] + H (x) = k n−1 ; n 2 Z k=1 Proof. Theorem 4.2. is proven by using the Principle of Mathematical Induction. Base Case: n = 1 x x (1) Y k Y k[P0((x−k)+1)] H (x) = k = k [Since P0(ξ) = 1; 8 ξ 2 N] k=1 k=1 Inductive Hypothesis: x Y H(m)(x) = kk[Pm−1((x−k)+1)] k=1 To prove: x Y H(m+1)(x) = kk[Pm((x−k)+1)] k=1 x Y H(m+1)(x) = H(m)(k) [From equation (4.1)] k=1 x " k # Y Y = ii[Pm−1((k−i)+1)] [From Inductive Hypothesis] k=1 i=1 1 2 x Y Y Y = ii[Pm−1((1−i)+1)] ii[Pm−1((2−i)+1)] ··· ii[Pm−1((x−i)+1)] i=1 i=1 i=1 h i h i = 11·Pm−1(1)11·Pm−1(2) ··· 11·Pm−1(x) 22·Pm−1(1)22·Pm−1(2) ··· 22·Pm−1(x−1) h i ··· xx·Pm−1(1) [On rearranging terms] 2 x 3 2x−1 3 2 1 3 X X X 1·6 7 2·6 7 x·6 7 4 Pm−1(k)5 4 Pm−1(k)5 4 Pm−1(k)5 = 1 k=1 2 k=1 ··· x k=1 = 11·Pm(x)22·Pm(x−1) ··· xx·Pm(1) [From equation 2.4] x Y = kk[Pm((x−k)+1)] k=1 6 VIGNESH RAMAN 5. Visual Exploration into Number Patterns Intuition behind the formulae of the superfactorial and hyperfactorial can be developed on observation of the number pattern they manifest themselves into (note: product of all numbers in a pattern is the output of the function). The below list will include some illustrations which will attempt to display the nature of the functions : (1) sf 1(5) =1 ∗ 2 ∗ 3 ∗ 4 ∗ 5 1 ∗ 2 ∗ 3 ∗ 4 1 ∗ 2 ∗ 3 1 ∗ 2 1 (2) H1(5) =5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 5 ∗ 4 ∗ 3 ∗ 2 5 ∗ 4 ∗ 3 5 ∗ 4 5 (3) sf 2(5) =1 ∗ 2 ∗ 3 ∗ 4 ∗ 5 1 ∗ 2 ∗ 3 ∗ 4 1 ∗ 2 ∗ 3 1 ∗ 2 1 1 ∗ 2 ∗ 3 ∗ 4 1 ∗ 2 ∗ 3 1 ∗ 2 1 1 ∗ 2 ∗ 3 1 ∗ 2 1 1 ∗ 2 1 1 (4) H2(5) =5 ∗ 4 ∗ 3 ∗ 2 ∗ 1 4 ∗ 3 ∗ 2 ∗ 1 3 ∗ 2 ∗ 1 2 ∗ 1 1 5 ∗ 4 ∗ 3 ∗ 2 4 ∗ 3 ∗ 2 3 ∗ 2 2 5 ∗ 4 ∗ 3 4 ∗ 3 3 5 ∗ 4 4 5 (5) M(8) =1 ∗ 2 ∗ 3 ∗ 4 ∗ 5 ∗ 6 ∗ 7 ∗ 8 1 ∗ 2 ∗ 3 6 ∗ 7 ∗ 8 1 ∗ 2 7 ∗ 8 1 8 THE GENERALIZED SUPERFACTORIAL, HYPERFACTORIAL AND PRIMORIAL FUNCTIONS7 (6) M(9) =1 ∗ 2 ∗ 3 ∗ 4 ∗ 5 ∗ 6 ∗ 7 ∗ 8 ∗ 9 1 ∗ 2 ∗ 3 ∗ 4 6 ∗ 7 ∗ 8 ∗ 9 1 ∗ 2 ∗ 3 7 ∗ 8 ∗ 9 1 ∗ 2 8 ∗ 9 1 9 (7) N(7) =1 ∗ 2 ∗ 3 ∗ 4 ∗ 5 ∗ 6 ∗ 7 2 ∗ 3 ∗ 4 ∗ 5 ∗ 6 3 ∗ 4 ∗ 5 4 (8) N(6) =1 ∗ 2 ∗ 3 ∗ 4 ∗ 5 ∗ 6 2 ∗ 3 ∗ 4 ∗ 5 3 ∗ 4 Item numbers 1 to 4 offers a visual insight into the pattern generated by the super- factorial and hyperfactorial (Degrees 1 and 2) functions.
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