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A78 Integers 18 (2018) #A78 INTEGERS 18 (2018) NEW CONGRUENCES AND FINITE DIFFERENCE EQUATIONS FOR GENERALIZED FACTORIAL FUNCTIONS Maxie D. Schmidt School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia [email protected], [email protected] Received: 2/11/17, Revised: 5/8/18, Accepted: 9/24/18, Published: 10/5/18 Abstract The generalized factorial product sequences we study in this article are defined symbolically by p (↵, R) = R(R + ↵) (R + (n 1)↵) for any indeterminate R. This definition of the generalized n · · · − factorial functions pn(↵, R) includes variants of the classical single and double factorial functions, multiple ↵-factorial functions, n!(↵), rising and falling factorial polynomials such as the Pochhammer symbol, and other well-known sequences as special cases. We use the rationality of the generalized th h convergent functions, Convh(↵, R; z), to the infinite J-fraction expansions enumerating these generalized factorial product sequences first proved by the author in 2017 to construct new con- gruences and h-order finite di↵erence equations for generalized factorial functions. Applications of the new results we prove in this article include new finite sums and congruences for the ↵-factorial functions, restatements of classical necessary and sufficient conditions of the primality of special integer subsequences and tuples, and new finite sums for the single and double factorial functions modulo integers h 2. ≥ 1. Notation and Other Conventions in the Article 1.1. Notation and Special Sequences Most of the special conventions in the article are consistent with the notation employed within the Concrete Mathematics reference, and the conventions defined in the introduction to the first articles [11, 12]. These conventions include particular notational variants for usage of Iverson’s convention, n n k n [i = j]δ δi,j, bracket notation for the Stirling numbers, k = ( 1) − s(n, k) and k = S(n, k), ⌘ (r−) n r (1) and notation for the generalized r-order harmonic numbers⇥ ⇤, Hn = k=1 k− where Hn = Hn denotes the sequence of first-order harmonic numbers. We will use the convention of denoting the P falling factorial function by xn = x!/(x n)!, the rising factorial function as xn = Γ(x + n)/Γ(x), − or equivalently by the Pochhammer symbol, (x) = x(x + 1)(x + 2) (x + n 1), when x is a n · · · − non-negative natural number. Within the article the notation g (n) g (n) (mod N , N , . , N ) 1 ⌘ 2 1 2 k is understood to mean that the congruence, g (n) g (n) (mod N ), holds modulo any of the 1 ⌘ 2 j bases, N , for 1 j k. Finally, we adopt the convention that the natural numbers are indexed j starting from zero: N := 0, 1, 2, 3, . { } INTEGERS: 18 (2018) 2 1.2. Mathematica Summary Notebook Document and Computational Reference Information The article is prepared with a more extensive set of computational data and software routines released as open source software to accompany the examples and numerous other applications suggested as topics for future research and investigation within the article. It is highly encouraged, and expected, that the interested reader obtain a copy of the summary notebook reference and computational documentation prepared in this format to assist with computations in a multitude of special case examples cited as particular applications of the new results. The prepared summary notebook file https://goo.gl/KK5rNnmultifact-cfracs-summary.nb , attached to this manuscript contains the working Mathematica code to verify the formulas, proposi- tions, and other identities cited within the article [13]. Given the length of this and the first article, the Mathematica summary notebook included with this submission is intended to help the reader with verifying and modifying the examples presented as applications of the new results cited below. The summary notebook also contains numerical data corresponding to computations of multiple examples and congruences specifically referenced in several places by the applications given in the next sections of the article. 2. Introduction 2.1. Motivation In this article, we extend the results from [11] providing infinite J-fraction expansions for the typically divergent ordinary generating functions (OGFs) of generalized factorial product sequences of the form p (↵, R) := R(R + ↵)(R + 2↵) (R + (n 1)↵) [n 1] + [n = 0] , (1) n ⇥ · · · ⇥ − ≥ δ δ when R depends linearly on n. Notable special cases of (1) that we are particularly interested in enumerating through the convergents to these J-fraction expansions include the multiple, or + ↵-factorial functions, n!(↵), defined for ↵ Z as 2 n (n ↵)! , if n > 0; · − (↵) n! = 1, if ↵ < n 0; (2) (↵) 8 − <>0, otherwise, > and the generalized factorial functions: of the form pn(↵, βn + γ) for ↵, β, γ Z, ↵ = 0, and β, γ 2 6 not both zero. The second class of special case products are related to the Gould polynomials, x x an n Gn(x; a, b) = x an −b , through the following identity ([12, 3.4.2],[10, 4.1.4]): − · § § ( ↵)n+1 p (↵, βn + γ) = − G (γ ↵ β; β, ↵) . (3) n γ ↵ β ⇥ n+1 − − − − − − + The ↵-factorial functions, (↵n d)!(↵) for ↵ Z and some 0 d < ↵, form special cases of (3) − 2 where, equivalently, (↵, β, γ) ( ↵, ↵, d) and (↵, β, γ) (↵, 0, ↵ d) [11, 6]. The ↵-factorial ⌘ − − ⌘ − § INTEGERS: 18 (2018) 3 n functions are expanded by the triangles of Stirling numbers of the first kind, k , and the ↵-factorial coefficients, n , respectively, in the following forms [5, 12]: k ↵ ⇥ ⇤ ⇥ ⇤ n n/↵ n m m + n!(↵) = d e ( ↵)d ↵ e n , n 1, ↵ Z (4) m − 8 ≥ 2 m=0 X n n 1+↵ n 1+↵ −↵ + 1 − m m + = b c ( 1)b ↵ c (n + 1) , n 1, ↵ Z m + 1 − 8 ≥ 2 m=0 ↵ X n n n m m 1 (↵n d)! = (↵ d) ( 1) − (↵n + 1 d) − − (↵) − ⇥ m − − m=1 ↵ Xn n + 1 n m m + = ( 1) − (↵n + 1 d) , n 1, ↵ Z , 0 d < ↵. m + 1 − − 8 ≥ 2 m=0 ↵ X A careful treatment of the polynomial expansions of these generalized ↵-factorial functions through the coefficient triangles in (4) is given in [12]. 2.2. Summary of the J-fraction Results For all h 2, we can generate the generalized factorial product sequences, p (↵, R), through ≥ n the strictly rational generating functions provided by the hth convergent functions, denoted by Convh (↵, R; z), to the infinite continued fraction series established by [11]. In particular, we have series expansions of these convergent functions proved in [11] given by 1 Convh (↵, R; z) := ↵R z2 1 R z · − · − 2↵(R + ↵) z2 1 (R + 2↵) z · − · − 3↵(R + 2↵) z2 1 (R + 4↵) z · − · − · · · 1 (R + 2(h 1)↵) z − − · FP (↵, R; z) = h (5) FQh(↵, R; z) h n 1 n = pn(↵, R)z + [pn(↵, R) (mod h)] z , n=0 X n>hX where the convergent function numerator and denominator polynomial subsequences which provide the characteristic expansions of (5) are given in closed-form by (6) and (7) below. For example, when h := 2 we have a convergent function approximation given by 1 (2↵ + R)z Conv (↵, R; z) = − 2 1 2(↵ + R)z + R(↵ + R)z2 − = 1 + Rz + R(↵ + R)z2 + R(↵ + R)(↵ + 2R)z3 + R(↵ + R)2(4↵ + R)z4 + R(↵ + R)2(R2 + 8↵R + 8↵2)z5 + . · · · INTEGERS: 18 (2018) 4 More generally, we have that h k 1 h − FQ (↵, R; z) = ( 1)k (R + (h 1 j)↵) zk (6) h k − 0 − − 1 j=0 kX=0 ✓ ◆ Y h @ A h R = + h k ( ↵z)k k ↵ − k − kX=0 ✓ ◆ ✓ ◆ h (R/↵ 1) 1 = ( ↵z) h! L − (↵z)− , − · ⇥ h (β) when Ln (x) denotes an associated Laguerre polynomial, and where h 1 − n FPh(↵, R; z) = Ch,n(↵, R)z (7a) n=0 X h 1 n − h i n = ( 1) pi ( ↵, R + (h 1)↵) pn i (↵, R) z (7b) i − − − − n=0 i=0 ! X X ✓ ◆ h 1 n − h n = (1 h R/↵)i (R/↵)n i (↵z) . (7c) i − − − n=0 i=0 ! X X ✓ ◆ The coefficients of the polynomial powers of z in the previous several expansions of (7), denoted by C (↵, R) := [zn] FP (↵, R; z) for 0 n < h, also have the following multiple, alternating sum h,n h expansions involving the Stirling number triangles [11, 5.2]: § m s h m k m n R R − s Ch,n(↵, R) = ( 1) ↵ 1 h (8a) k s m − ↵ n k ↵ − ! ⇥ 0 m k n ✓ ◆✓ ◆ ✓ ◆ − ✓ ◆ 0 Xs n h m k n k m n s m t s = − ( 1) ↵ − (h 1) − R (8b) k t m s t − − ⇥ 0 m k n ✓✓ ◆✓ ◆ − ◆ 0 Xt s n m s h h m k s R R − = ( 1)m↵n 1 i! (8c) k i s m i − ↵ n k ↵ − ⇥ 0 m k n ✓ ◆✓ ◆✓ ◆ ⇢ ✓ ◆ − ✓ ◆ 0 Xi s n h m i h + v k s m+i v n = ( 1) − ↵ (8d) k s v v m i − ⇥ 0 m k n ✓ ◆✓ ◆✓ ◆✓ ◆ ⇢ 0 v Xi s n m s R R − 1 i! ⇥ ↵ n k ↵ − ⇥ ✓ ◆ − ✓ ◆ n h m k n k s m+s i m t n R = 0 − ( 1) − (h 1) − 1 ↵ . k t m s t i − − ⇥ ↵ i=0 0 m k n ✓ ◆✓ ◆ − ⇢ ✓ ◆i X B 0 Xt s n C B C @ A polynomial function of h only | {z } (8e) INTEGERS: 18 (2018) 5 th Given that the non-zero h convergent functions, Convh (↵, R; z), are rational in each of z, ↵, R for all h 1, and that the convergent denominator sequences, FQ (↵, R; z), have characteristic ≥ h expansions by the Laguerre polynomials and the confluent hypergeometric functions, we may ex- pand both exact finite sums and congruences modulo h↵t for the generalized factorial functions, pn(↵, βn + γ), by the distinct special zeros of these functions according to the following identities: n p (↵, R) = c (↵, R) ` (↵, R)n (9) n n,j ⇥ n,j j=1 X h p (↵, R) c (↵, R) ` (↵, R)n (mod h) n ⌘ h,j ⇥ h,j j=1 X n n 1 − n! = c ( ↵, n) ` ( ↵, n)b ↵ c (↵) n,j − ⇥ n,j − j=1 X h n 1 − h n! c ( ↵, n) ` ( ↵, n)b ↵ c (mod h, h↵, , h↵ ).
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