Explicit Expressions for the Related Numbers of Higher Order Appell Polynomials

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∞ tezt ezt tn = = B (z) . (1) et − 1 et−1 n n! t n=0 X These numbers and polynomials have a long history, which arise from Bernoulli’s calculations of power sums in 1713, that is,

m B (m + 1) − B jn = n+1 n+1 n + 1 j X=1 (see [23, p.5, (2.2)]). They have many applications in modern number theory, such as modular forms and Iwasawa theory [15]. For r ∈ N, in 1924, N¨orlund [21] generalized (1) to give a deﬁnition of higher order Bernoulli polynomials and numbers

∞ t r ezt tn ezt = = B(r)(z) . et − 1 et−1 r n n! t n=0 X We also have a similar expression of multiple power sums

m−1 k (t + l1 + · · · + ln) l1,...,lXn=0 in terms of higher order Bernoulli polynomials (see [19, Lemma 2.1]). The Euler polynomials are deﬁned by the generating function

∞ 2ezt ezt tn = = En(z) . et + 1 et+1 n! 2 n=0 X These polynomials were introduced by Euler who studied the alternating power sums, that is,

m (−1)mE (m +1)+ E (0) (−1)j+1jn = − n n 2 j X=1 (see [23, p.5, (2.3)]). We may also deﬁne the higher order Euler polynomial as follows ∞ 2 r ezt tn ezt = = E(r)(z) . et + 1 et+1 r n n! 2 n=0 X 2 In 1880, Appell [1] found that the Bernoulli and Euler polynomials are the sequences (An(z))n≥0 in the polynomial ring C[z], which satisfy the ′ recurrence relation An(z) = nAn−1(z) for n ≥ 1 and A0(z) is a non-zero constant polynomial. And the sequence (An(z))n≥0, now known as Appell polynomials, can also be deﬁned equivalently by using generating functions, ∞ tn C that is, let S(t)= n=0 an n! be a formal power series in [[t]] and a0 6= 0, we have ∞ P tn A (z) = S(t)ezt. (2) n n! n=0 X Recently, Bencherif, Benzaghou and Zerroukhat [4] established an identity for some Appell polynomials generalizing explicit formulas for generalized Bernoulli numbers and polynomials. Since a0 6= 0, there exists the formal power series (for some dn ∈ C)

∞ 1 tn f(t)= = d (3) S(t) n n! n X=0 in C[[t]], and (2) becomes

∞ ezt tn = An(z) . (4) f(t) n! n=0 X We can also define the higher order Appell polynomials by the generating function ∞ ezt tn = A(r)(z) (5) (f(t))r n n! n X=0 (r) (r) (see [4, Théorème 1.1]). As in the classical case, we call an = An (0) the related numbers of higher order Appell polynomials, that is,

∞ 1 tn = a(r) (6) (f(t))r n n! n=0 X (1) and an = an the related numbers of Appell polynomials. In what follows, we assume d0 = 1 to normalize the expansion of f(t) in (3). The above deﬁnitions unify several generalizations of Bernoulli numbers and polynomials. For example,

3 1. The higher order generalized hypergeometric Bernoulli numbers and polynomials. Let (x)(n) = x(x + 1) . . . (x + n − 1) (n ≥ 1) with (x)(0) = 1 be the rising factorial. For positive integers M, N and r, put

∞ (M)(n) tn f(t)= F (M; M + N; t)= , 1 1 (n) n! n (M + N) X=0 the conﬂuent hypergeometric function (see [7]), in (2), we obtain the (r) higher order generalized hypergeometric Bernoulli polynomials BM,N,n(z), that is, ∞ ezt tn = B(r) (z) . r M,N,n n! 1F1(M; M + N; t) n=0 X (r) (r) When z = 0, BM,N,n = BM,N,n(0) are the higher order generalized hypergeometric Bernoulli numbers. When M = 1, the higher order (r) (r) hypergeometric Bernoulli polynomials BN,n(z)= B1,N,n(z) are studied by Hu and Kim in [13]. When r = M = 1, we have BN,n(z) = (1) B1,N,n(z) and

∞ ezt tN ezt/N! tn = t = BN,n(z) , (7) F (1; 1 + N; t) e − TN− (t) n! 1 1 1 n=0 X which is the hypergeometric Bernoulli polynomials deﬁned by Howard in [11, 12] and was studied by Kamano in [16]. From (7), if r = (1) M = N = 1, we have Bn(z)= B1,1,n(z), which reduces to the classical Bernoulli polynomials (1).

2. The higher order generalized hypergeometric Cauchy numbers. For positive integers M, N and r, put

∞ (M)(n)(N)(n) (−t)n f(t)= 2F1(M,N; N + 1; −t)= , (N + 1)(n) n! n=0 X the Gauss hypergeometric function, in (6), we obtain the higher order (r) generalized hypergeometric Cauchy numbers cM,N,n, that is,

∞ 1 tn = c(r) (8) r M,N,n n! 2F1(M,N; N + 1; −t) n=0 X 4 which has been studied by Komatsu and Yuan in [18]. When r = 1, (1) cM,N,n = cM,N,n are the generalized hypergeometric Cauchy numbers which has been studied by Komatsu in [17]. When r = M = N = 1, (1) cn = c1,1,n are classical Cauchy numbers (see [18, p. 383. Eq. (1)]), which can also be deﬁned by the well-known generating function

∞ t tn = c (9) log(1 + t) n n! n X=0 (see [26]).

In 1875, Glaisher stated the following classical determinant expression of Bernoulli numbers Bn in an article ([8, p.53]):

1 2! 1 1 1 3! 2! n . . . Bn = (−1) n! . . .. 1 . (10)

1 1 · · · 1 1 n! (n−1)! 2! 1 1 1 1 (n+1)! n! · · · 3! 2!

In 2010, Costabile and Longo [5, Theorem 2] obtained a determinant expression of the ﬁrst order Appell polynomials (4), and their result reduces to a diﬀerent determinant expression for Bernoulli numbers:

1 1 1 1 1 2 3 4 · · · n n+1 1 1 1 · · · 1 1

n 0 2 3 · · · n − 1 n (−1) Bn = 0 2 3 · · · n−1 n (n − 1)! 2 2 2 ...... 0 0 0 · · · n−1 n n−2 n−2

for n ≥ 1 (also see [6, p.7]). In 2016, from the integral expression for the generating function of Bernoulli polynomials, Qi and Chapman [22, Theorems 1 and 2] got two closed forms for Bernoulli polynomials, which provided an explicit formula for computing these special polynomials in terms of Stirling numbers of the second kind S(n, k). According to Wiki [25] (also see [22]), “In mathematics, a closed-form expression is a mathematical expression that can be evaluated in a ﬁnite number of operations. It may contain constants, variables, certain ‘well-known’ operations (e.g., + − ×÷), and functions (e.g., nth root,

5 exponent, logarithm, trigonometric functions, and inverse hyperbolic functions), but usually no limit.” It needs to mention that [22, Theorem 2] also shows a new determinant expression of Bernoulli polynomials, which reduces to another determinant expression of Bernoulli numbers. Recently, by di- rectly applying the generating functions instead of their integral expressions, Hu and Kim [14] obtained new closed form and determinant expressions of Apostol-Bernoulli polynomials [2, 3, 20]. In 2017, applying the Hasse-Teichmüller derivatives [9], Komatsu and Yuan [18] presented a determinant expression of the higher order generalized hypergeometric Cauchy numbers defined as in (8) (see [18, Theorem 4]). When r = M = N = 1, their formula recovers the classical determinant expression of Cauchy numbers cn (9) which was also stated in the article of Glaisher ([8, p.50]). In this note, generalizing the methods in [18], by using the Hasse-Teichmüller derivatives [9] (see Sec.2 below for a brief review), we shall obtain two explicit expressions for the related numbers of higher order Appell polynomials (Theorems 1 and 2). Theorem 2 gives a determinant expression for the related numbers of higher order Appell polynomials defined as in (6), which involves several determinant expressions of special numbers, such as the higher order generalized hypergeometric Bernoulli and Cauchy numbers stated above, thus recovers the classical determinant expressions of Bernoulli and Cauchy numbers (see the last section).

2 Hasse-Teichm¨uller derivatives

We shall introduce the Hasse-Teichmüller derivatives in order to prove our results. Let F be a field of any characteristic, F[[z]] the ring of formal power series in one variable z, and F((z)) the field of Laurent series in z. Let n be a nonnegative integer. We define the Hasse-Teichmüller derivative H(n) of order n by ∞ ∞ m H(n) c zm = c zm−n m m n m R ! m R X= X= ∞ m F F for m=R cmz ∈ ((z)), where R is an integer and cm ∈ for any m ≥ R. Note that m =0 if m

6 Lemma 1 ([24, 18]). For fi ∈ F[[z]] (i = 1,...,k) with k ≥ 2 and for n ≥ 1, we have (n) (i1) (ik) H (f1 · · · fk)= H (f1) · · · H (fk) . i1+···+ik=n i1,...,iXk≥0

The quotient rules can be described as follows. Lemma 2 ([9, 18]). For f ∈ F[[z]]\{0} and n ≥ 1, we have n 1 (−1)k H(n) = H(i1)(f) · · · H(ik)(f) (11) f f k+1 k=1 i1+···+ik=n X i1,...,iXk≥1 n n + 1 (−1)k = H(i1)(f) · · · H(ik)(f) . (12) k + 1 f k+1 k=1 i1+···+ik=n X i1,...,iXk≥0

3 Results and their proofs

Recall (5) and (3) with d0 = 1 for a normalization. We have the following proposition on the recurrence for the related numbers of higher order Appell polynomials. Proposition 1. For n ≥ 1, we have n (r) d · · · d am i1 ir = 0 . i1! · · · ir! m! m=0 i1+···+ir=n−m X i1,...,iXr≥0

Proof. From (5) and (3), we have ∞ r ∞ tl tm 1= d a(r) l l! m m! ! m ! Xl=0 X=0 ∞ l ∞ m l! t (r) t =  di1 · · · dir  am i1! · · · ir! l! m! l=0 i1+···+ir=l m=0 ! X i1,...,iXr≥0  X ∞ n  n n (n − m)!di1 · · · dir (r) t = am . m i1! · · · ir! n! n=0 m=0 i1+···+ir=n−m X X i1,...,iXr≥0 Comparing the coeﬃcients of tn in the above equality, we get our result.

7 By using Proposition 1, we have Corollary 1.

n−1 (r) (r) di1 · · · dir am an = −n! (13) i1! · · · ir! m! m=0 i1+···+ir=n−m X i1,...,iXr≥0

(r) with a0 = 1. We have an explicit expression for the related numbers of higher order (r) Appell polynomials an . Theorem 1. For n ≥ 1, we have

n (r) k an = n! (−1) Dr(e1) · · · Dr(ek) , k=1 e1+···+ek=n X e1,...,eXk≥1 where di1 · · · dir Dr(e)= . (14) i1! · · · ir! i1+···+ir=e i1,...,iXr≥0

Proof. We make an application of the Hasse-Teichm¨uller derivatives which were introduced in Sec.2 above. r Put h(t)= f(t) , where

∞ tj f(t)= d . j j! j X=0 Since

∞ d j H(i)(f) = j tj−i t=0 j! i j=i t=0 X di = i! by the product rule of the Hasse-Teichm¨uller derivative in Lemma 1, we get

H(e)(h) = H(i1)(f) · · · H(ir)(f) t=0 t=0 t=0 i1+···+ir=e i1,...,iXr≥0

8 di1 · · · dir = := Dr(e) . i1! · · · ir! i1+···+ir=e i1,...,iXr≥0

Hence, by the quotient rule of the Hasse-Teichm¨uller derivative in Lemma 2 (11), we have

(r) n k an (−1) (e1) (ek) = k+1 H (h) · · · H (h) n! h t t=0 t=0 k=1 =0 e1+···+ek=n X e1,...,eXk≥1

n k = (−1) Dr(e1) · · · Dr(ek) . k=1 e1+···+ek=n X e1,...,eXk≥1

(r) Now, we show a determinant expression of an .

Theorem 2. For n ≥ 1, we have

Dr(1) 1 D (2) D (1) r r (r) n . . .. an = (−1) n! . . . 1 .

Dr(n − 1) Dr(n − 2) · · · Dr(1) 1

D (n) D (n − 1) · · · D (2) D (1) r r r r

where Dr(e) are given in (14). Proof. The proof is an application of the inductive method. n (r) (r) (−1) an For simplicity, put An = n! . Then, we shall prove that for any n ≥ 1

Dr(1) 1 D (2) D (1) r r (r) . . . An = . . .. 1 . (15)

Dr(n − 1) Dr(n − 2) · · · Dr(1) 1

D (n) D (n − 1) · · · D (2) D (1) r r r r

When n = 1, (15) is valid, because by Theorem 1

(r) Dr(1) = d1 = A1 .

9 Assume that (15) is valid up to n − 1. Notice that by (13), we have

n (r) l−1 (r) An = (−1) An−lDr(l) . Xl=1 Thus, by expanding the ﬁrst row of the right-hand side (15), it is equal to

Dr(2) 1 D (3) D (1) r r (r) . . .. Dr(1)An−1 − . . . 1

Dr(n − 1) Dr(n − 3) · · · Dr(1) 1

D (n) D (n − 2) · · · D (2) D (1) r r r r (r) (r) = D (1)A − D (2)A r n−1 r n−2 Dr(3) 1 D (4) D (1) r r . . . + . . .. 1

Dr(n − 1) Dr(n − 4) · · · Dr(1) 1

D (n) D (n − 3) · · · D (2) D (1) r r r r

r r D (n − 1) 1 = D (1) A( ) − D (2)A( ) + · · · + (−1)n−2 r r n−1 r n−2 D (n ) D (1) r r n l−1 (r) (r) = (−1) Dr(l)An−l = An . Xl=1 (r) (r) Note that A1 = Dr(1) and A0 = 1. When r = 1, we have the following determinant expression for the related numbers of Appell polynomials.

Theorem 3. For n ≥ 1, we have

d1 1 d2 d 2! 1 n . . .. an = (−1) n! . . . 1 .

dn−1 dn−2 · · · d 1 (n−1)! (n−2)! 1 dn dn−1 d2 · · · d1 n! (n−1)! 2!

10 4 Applications of Theorem 2

In this section, as applications of Theorem 2, we show several determinant expressions of special numbers, including the higher order generalized hypergeometric Bernoulli and Cauchy numbers introduced in Sec.1, thus recovers the classical determinant expressions of Bernoulli and Cauchy numbers. 1. The generalized hypergeometric Bernoulli numbers. For positive integers M, N, put ∞ (M)(n) tn f(t)= F (M; M + N; t)= , 1 1 (n) n! n (M + N) X=0 in Theorem 3, we obtain the determinant expression of the generalized hypergeometric Bernoulli numbers: n BM,N,n = (−1) n! (M)(1) (M+N)(1) 1 (M)(2) (M)(1) (2) (1) 2!(M+N) (M+N) . . . × . . .. 1 . (n−1) (n−2) (1) (M) (M) · · · (M) 1 (n−1)!(M+N)(n−1) (n−2)!(M+N)(n−2) (M+N)(1) (n) (n−1) (2) (1) (M) (M) · · · (M) (M) n!(M+N)(n) (n−1)!(M+N)(n−1) 2!(M+N)(2) (M+N)(1)

Letting M = N = 1 in the above, we obtain Glaisher’s determinant expression of Bernoulli numbers (10). 2. The generalized hypergeometric Cauchy numbers. For positive integers M, N, put ∞ (M)(n)(N)(n) (−t)n f(t)= F (M,N; N + 1; −t)= , 2 1 (n) n! n (N + 1) X=0 in Theorem 3, we obtain the determinant expression of the generalized hypergeometric Cauchy numbers: M·N N+1 1 (M)(2)N M·N 2!(N+2) N+1

. . .. cM,N,n = n! . . . 1 (n−1) (n−2) (M) N (M) N · · · M·N 1 (n−1)!(N+n−1) (n−2)!(N+n−2) N+1 (M)(n)N (M)(n−1)N (M)(2)N · · · M·N n!(N+n) (n−1)!(N+n−1) 2!(N+2) N+1

11 (also see [18, Theorem 3]). Letting M = N = 1 in the above, we recover the classical determinant expression of Cauchy numbers ([8, p.50]):

1 2 1 1 1 3 2 . . .. cn = n! . . . 1 .

1 1 · · · 1 1 n n−1 2 1 1 · · · 1 1 n+1 n 3 2

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