Zeros of hypergeometric functions in the p-adic setting

Abstract

McCarthy (Pac J Math 261(1):219–236, 2013) defined hypergeometric functions in the p-adic setting over finite fields using p-adic gamma functions. These functions possess many properties that are analogous to classical hypergeometric type identities. In this paper, we investigate values of two generic families of these hypergeometric functions that we denote by ${_nG_n}(t)_p$ and ${_n{\widetilde{G}}_n}(t)_p$ for $n\ge 3$, and $t\in {\mathbb {F}}_p$, the finite field with p elements. These results generalize special cases of p-adic analogues of Whipple’s theorem and Dixon’s theorem of classical hypergeometric series. We also examine zeros of the functions ${_nG_n}(t)_p$, and ${_n{\widetilde{G}}_n}(t)_p$ over ${\mathbb {F}}_p$. Moreover, we classify the values of t for which ${_nG_n}(t)_p=0$ for infinitely many primes. For example, we show that there are infinitely many primes for which ${_{2k}G_{2k}}(-1)_p=0$. In contrast, for $t\ne 0$ there are no primes for which ${_{2k}{\widetilde{G}}_{2k}}(t)_p=0$.

1 Introduction

McCarthy [18] introduced a function in terms of quotients of p-adic gamma functions that can be understood as p-adic analogue of classical hypergeometric series. He developed these functions to generalize Greene’s hypergeometric functions over finite fields to wider classes of primes. In this paper, we aim to investigate the values of certain families of these hypergeometric functions. To be specific, for a positive integer $n\ge 3$, we consider two families of hypergeometric functions in the p-adic setting with n pairs of arbitrary parameters essentially depend on n and determine all possible values and explore their consequences.

For a complex number a and integer $k\ge 0$, the rising factorial denoted by $(a)_k$ is defined by $(a)_k:=a(a+1)(a+2)\cdots (a+k-1)$ for $k>0$ and $(a)_0:=1.$ If $a_i,b_i,\lambda \in {\mathbb {C}}$ with $b_i\not \in \{\ldots ,-3,-2,-1,0\},$ then the classical hypergeometric series ${_{r+1}F_r}(\lambda )$ is defined by

$$\begin{aligned} {_{r+1}}F_{r}\left( \begin{array}{cccc} a_1, &{} a_2, &{} \ldots , &{} a_{r+1} \\ &{} b_1, &{} \ldots , &{} b_r \end{array}\mid \lambda \right) :=\sum _{k=0}^{\infty }\frac{(a_1)_k\cdots (a_{r+1})_k}{(b_1)_k\cdots (b_r)_k}\cdot \frac{\lambda ^k}{k!}. \end{aligned}$$

In this series, if one of the numerator parameters is equal to a nonpositive integer, for example let $a_1=-n$, where $n\in {\mathbb {N}}\cup \{0\}$, then the series terminates and the function is a polynomial of degree n in $\lambda$. The problem of describing the zeros of the polynomials ${_2F_1}\left( \begin{array}{cc} -n, &{} b \\ &{} c \end{array}\mid z\right)$ when b and c are complex arbitrary parameters is of particular interest. Indeed it has not been explored completely, and even when b, c are real. The zero location of special classes of such polynomials with restrictions on the parameters b and c can be found in [1, 3,4,5, 7, 8] and the asymptotic zero distribution of certain classes have been investigated in [2, 6, 9, 10, 22]. In view of these results, it is natural to study zeros of other hypergeometric type functions and investigate their consequences. In this paper, we study zeros of p-adic analogues of hypergeometric functions from a number theoretic point of view.

Let $\Gamma _p(\cdot )$ denote the Morita’s p-adic gamma function and $\omega$ denote the Teichmüller character of ${\mathbb {F}}_p$ satisfying $\omega (a)\equiv a\pmod {p}$. Let ${\overline{\omega }}$ denote the character inverse of $\omega$. For $x\in {\mathbb {Q}}$ let $\lfloor x\rfloor$ denote the greatest integer less than or equal to x and $\langle x\rangle$ denote the fractional part of x, satisfying $0\le \langle x\rangle <1$. Using these notations, we recall the definition of hypergeometric function in the p-adic setting.

Definition 1.1

[18, Definition 5.1] Let p be an odd prime and $t \in {\mathbb {F}}_p$. For positive integer n and $1\le k\le n$, let $a_k$, $b_k$ $\in {\mathbb {Q}}\cap {\mathbb {Z}}_p$. Then

$$\begin{aligned}&{_n{\mathbb {G}}_n}\left[ \begin{array}{cccc} a_1, &{} a_2, &{} \ldots , &{} a_n \\ b_1, &{} b_2, &{} \ldots , &{} b_n \end{array}\mid t \right] _p\\&\quad :=\frac{-1}{p-1}\sum _{a=0}^{p-2}(-1)^{an}~~{\overline{\omega }}^a(t)\\&\qquad \times \prod \limits _{k=1}^n(-p)^{-\lfloor \langle a_k \rangle -\frac{a}{p-1} \rfloor -\lfloor \langle -b_k \rangle +\frac{a}{p-1}\rfloor } \frac{\Gamma _p(\langle a_k-\frac{a}{p-1}\rangle )}{\Gamma _p(\langle a_k \rangle )} \frac{\Gamma _p(\langle -b_k+\frac{a}{p-1} \rangle )}{\Gamma _p(\langle -b_k \rangle )}. \end{aligned}$$

This function is also known as p-adic hypergeometric function. It is important to note that the value of this function depends only on the fractional part of the parameters $a_k$ and $b_k$. Therefore, we may assume that $0\le a_k,b_k<1.$

Here, we study certain two families of these hypergeometric functions. Namely, we consider

$$\begin{aligned} {_nG_n}(t)_p:&={_n{\mathbb {G}}_n}\left[ \begin{array}{ccccc} \frac{1}{2n}, &{} \frac{3}{2n}, &{} \frac{5}{2n}, &{} \ldots , &{} \frac{2n-1}{2n}\\ 0, &{} \frac{1}{n}, &{} \frac{2}{n}, &{} \ldots , &{} \frac{n-1}{n} \end{array}\mid t\right] _p, \end{aligned}$$

and

$$\begin{aligned} {_n{\widetilde{G}}_n}(t)_p:&={_n{\mathbb {G}}_n}\left[ \begin{array}{ccccc} \frac{1}{2}, &{} \frac{1}{2(n-1)}, &{} \frac{3}{2(n-1)}, &{} \ldots , &{} \frac{2n-3}{2(n-1)} \\ 0, &{} \frac{1}{n}, &{} \frac{2}{n}, &{} \ldots , &{} \frac{n-1}{n} \end{array}\mid t\right] _p. \end{aligned}$$

It is well known that the classical hypergeometric series possess many powerful identities. For instance, a theorem of Gauss provides a special value of a general ${_2F_1}$-hypergeometric series. To be specific, Gauss established that

$$\begin{aligned} {_2F_1}\left( \begin{array}{cc} a, &{} b \\ &{} c \end{array}\mid 1\right) =\frac{\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}, \end{aligned}$$

provided $R(c-a-b)>0$. There are other major summation theorems of classical hypergeometric series including Dixon’s theorem [21, p.51]

$$\begin{aligned} {_3F_2}&\left( \begin{array}{ccc} a, &{} b, &{} c\\ ~&{} 1+a-b, &{} 1+a-c \end{array}\mid 1\right) \nonumber \\&=\frac{\Gamma (1+\frac{a}{2})\Gamma (1+a-b)\Gamma (1+a-c)\Gamma (1+\frac{a}{2}-b-c)}{\Gamma (1+a)\Gamma (1+\frac{a}{2}-b)\Gamma (1+\frac{a}{2}-c)\Gamma (1+a-b-c)}, \end{aligned}$$

(1.1)

and Whipple’s theorem [17, p. 54]

$$\begin{aligned} {_3F_2}&\left( \begin{array}{ccc} a, &{} 1-a, &{} c\\ ~&{} b, &{} 1-b+2c \end{array}\mid 1\right) \nonumber \\&=\frac{\pi \Gamma (b)\Gamma (1-b+2c)}{\Gamma (\frac{1}{2}(a+b))\Gamma (\frac{1}{2}(1+a-b+2c))\Gamma (\frac{1}{2}(1-a+b))\Gamma (\frac{1}{2}(2-a-b+2c))}. \end{aligned}$$

(1.2)

In view of this examples, it is natural to seek their p-adic analogues and develop further identities of p-adic hypergeometric functions. In Sect. 2, we discuss the values of ${_nG_n}(t)_p.$ By investigating this values, we obtain their zeros. In Sect. 3 we explore the values of ${_n{\widetilde{G}}_n}(t)_p$ and examine their zeros. Most of the results are deduced by simplifying certain character sums and expressing the characters sums as hypergeometric functions in the p-adic setting. Section 4 includes the basic properties of characters, Gauss sums, Jacobi sums, and p-adic gamma functions we need. In Sect. 4, we also state the Hasse-Davenport theorem and the Gross–Koblitz formula which we use several times. Section 5 is devoted to the proofs of the main results that are stated in Sects. 2 and 3.

2 Values of ${_nG_n}(t)_p$

As our first result, we obtain the values of ${_nG_n}(t)_p$ in the following theorem. Let $\varphi (\cdot ):=\genfrac(){}{}{\cdot }{p}$ be the Legendre symbol.

Theorem 2.1

Let $n\ge 3$ be a fixed positive integer and p be an odd prime such that $p\not \mid n$. Let $d=\gcd {(n,p-1)}$ and $t\in {\mathbb {F}}_p$. Then the following are true.

• (1) If $t^{\frac{p-1}{d}}\equiv 1\pmod {p}$, then ${_nG_n}(t)_p=\displaystyle \sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ a^n\equiv t\pmod {p} \end{array}}\varphi (a)\varphi (a-1)$.

• (2) If $t^{\frac{p-1}{d}}\not \equiv 1\pmod {p}$, then ${_nG_n}(t)_p=0$.

There are two questions that emerge from Theorem 2.1. The first concerns the investigation of zeros of the p-adic hypergeometric function ${_nG_n}(t)_p$ over ${\mathbb {F}}_p$. Moreover, if $x\in {\mathbb {Q}}$ is a zero of ${_nG_n}(t)_p$ over ${\mathbb {F}}_p$, then the second question is whether x is a zero of ${_nG_n}(t)_p$ for infinitely many primes p. We first discuss a very special case which illustrates the zeros of the function: ${_3G_3}(t)_p={_3{\mathbb {G}}_3}\left[ \begin{array}{ccc} \frac{1}{2}, &{} \frac{1}{6}, &{} \frac{5}{6}\\ 0, &{} \frac{1}{3}, &{} \frac{2}{3} \end{array}\mid t\right] _p$.

The more general case will be discussed in Corollary 2.5.

Theorem 2.2

If $p>3$ is a prime, then the following are true for $t\in {\mathbb {F}}_p^{\times }.$

• (1) Suppose that $p\equiv 1\pmod {3}$ and g is a primitive root modulo p. If $t\ne 1$ then

  $$\begin{aligned} {_3G_3}(t)_p=0 \end{aligned}$$

  if and only if $t=g^i$ with $\gcd {(i,3)}=1.$ Moreover, if $t=1$ then ${_3G_3(1)}_p=0$ if and only if $p\equiv 7\pmod {12}$.

• (2) Suppose that $p\not \equiv 1\pmod {3}.$ Then ${_3G_3}(t)_p=0$ if and only if $t=0,1.$

Remark

It is important to note that ${_3G_3}(1)_p=0$ for infinitely many primes p. If $g\in {\mathbb {Z}}$ is a primitive root modulo p for infinitely many primes of the form $3k+1$, then ${_3G_3}(g)_p=0$ for infinitely many primes $p\equiv 1\pmod 3$. The existence of such kind of primitive roots is certainly related to Artin’s conjecture on primitive roots.

If we put $a=\frac{1}{2}$, $b=\frac{1}{6}$, and $c=\frac{5}{6}$ in (1.1), we obtain

$$\begin{aligned} {_3F_2}&\left( \begin{array}{ccc} \frac{1}{2}, &{} \frac{1}{6}, &{} \frac{5}{6}\\ &{} 1+\frac{1}{3}, &{} \frac{2}{3} \end{array}\mid 1\right) =\frac{\Gamma (1+\frac{1}{4})\Gamma (1+\frac{1}{3})\Gamma (\frac{2}{3})\Gamma (\frac{1}{4})}{\Gamma (1+\frac{1}{2})\Gamma (1+\frac{1}{12})\Gamma (\frac{5}{12})\Gamma (\frac{1}{2})}. \end{aligned}$$

(2.1)

This is a particular case of Dixon’s theorem (1.1) and gives motivation to present a p-adic analogue of (2.1) in the following corollary.

Corollary 2.3

If $p>3$ is a prime and $\alpha _p^2=-3$ for primes $p\equiv 1\pmod {12}$, then we have

$$\begin{aligned} {_3G_3}(1)_p=\left\{ \begin{array}{ll} 2\cdot \genfrac(){}{}{\alpha _p}{p}, &{} \hbox {{if}}~{p}\equiv 1\pmod {12};\\ 0, &{} \hbox {{if}}~{p}\not \equiv 1\pmod {12}. \end{array}\right. \end{aligned}$$

Moreover, if we put $n=3$ in Theorem 2.1, then for $t\in {\mathbb {F}}_p$, we obtain the values of ${_3G_3}(t)_p$ for all primes $p>3$. To be specific, we obtain

Corollary 2.4

• (1) Let $p\equiv 1\pmod {3}$ and $t\ne 0,1$. If $t^{\frac{p-1}{3}}\equiv 1\pmod {p}$, then

  $$\begin{aligned} {_3G_3}(t)_p=\displaystyle \sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ a^3\equiv t\pmod {p} \end{array}}\varphi (a(a-1)), \end{aligned}$$

  and if $t^{\frac{p-1}{3}}\not \equiv 1\pmod {p}$ then ${_3G_3}(t)_p=0$.

• (2) Let $p\equiv 2\pmod {3}$ and $t\ne 0,1$. Then ${_3G_3}(t)_p=\varphi (t^{\frac{2p-1}{3}})\varphi (t^{\frac{2p-1}{3}}-1)$.

The above corollary gives an extension of Corollary 2.3 over ${\mathbb {F}}_p$ and can be understood as some kind of extension over ${\mathbb {F}}_p$ of the particular case (2.1) of Dixon’s theorem in the p-adic setting.

In Theorem 2.1, we show that if $t^{\frac{p-1}{d}}\not \equiv 1\pmod {p}$ then ${_nG_n}(t)_p=0$ for all primes $p\not \mid n$, where $d=\gcd {(n,p-1)}$. More generally, it is interesting to examine the zeros of ${_nG_n}(t)_p.$

Corollary 2.5

Let $n\ge 3$ be a positive integer and p be an odd prime such that $p\not \mid n$. If $d=\gcd {(n,p-1)},$ then the following are true for $t^{\frac{p-1}{d}}\equiv 1\pmod {p}.$

• (1) Let n be even. If $t\ne 1$ and $a_1,a_2,\ldots ,a_d$ are the incongruent solutions of $y^n\equiv t\pmod {p}$ and $\displaystyle \sum _{i=1}^{d}\varphi (a_i(a_i-1))=0$, then ${_nG_n}(t)_p=0$. On the other hand, if $t=1$, then ${_nG_n}(1)_p\ne 0$.

• (2) Let n be odd. If $t=1$ and $a_1,a_2,\ldots ,a_{d-1}\ne 1$ are the incongruent solutions of the congruence $y^n\equiv 1\pmod {p}$ such that $\displaystyle \sum _{i=1}^{d}\varphi (a_i(a_i-1))=0$, then ${_nG_n}(1)_p=0$. On the other hand, if $t\ne 1$, then ${_nG_n}(t)_p\ne 0$.

The following corollaries provide particular values of $t\in {\mathbb {Q}}$ such that ${_nG_n}(t)_p=0$ for infinitely many primes p.

Corollary 2.6

Let $n\ge 3$ be an odd integer. Then ${_nG_n}(1)_p=0$ for infinitely many primes p such that $\gcd {(n,p(p-1))}=1$. If $t\ne 0,1$, then ${_nG_n}(t)_p=\varphi \left( \frac{a}{a-1}\right) \ne 0$ for all primes p such that $\gcd {(n,p(p-1))}=1$, where a is the unique solution of $y^n\equiv t\pmod {p}.$

Corollary 2.7

Let $n\ge 4$ be an even integer. Then ${_nG_n}(-1)_p=0$ for infinitely many primes p such that $p\equiv 3\pmod {4}.$

3 Values of ${_n{\widetilde{G}}_n}(t)_p$

In this section, we explore the values of the function ${_n{\widetilde{G}}_n}(t)_p$. We express these values in terms of roots of certain polynomial over ${\mathbb {F}}_p$.

Theorem 3.1

Let $n\ge 3$ be an integer and p be an odd prime such that $p\not \mid n(n-1)$. For $t\in {\mathbb {F}}_p^\times$ let $f_t(y):=y^n-y^{n-1}+\frac{(n-1)^{n-1}t}{n^n}$ be a polynomial over ${\mathbb {F}}_p$, and

$$\begin{aligned} \beta _n(t):=\left\{ \begin{array}{ll} 1, &{}\hbox {if} ~n ~\hbox {is odd;}\\ 1-(p-1)\varphi ((1-n)t), &{} \hbox {if} ~n \hbox {~is even.} \end{array} \right. \end{aligned}$$

Then we have

$$\begin{aligned}&{_n{\widetilde{G}}_n}(t)_p =\frac{\beta _n(t)-1}{p}+\sum _{f_t(a)\equiv 0\pmod {p}}\varphi (a(a-1)). \end{aligned}$$

If we put $a=\frac{1}{4}$, $b=\frac{1}{3}$, and $c=\frac{1}{2}$ in (1.2), then we obtain

$$\begin{aligned} {_3F_2}\left( \begin{array}{ccc} \frac{1}{4}, &{} \frac{3}{4}, &{} \frac{1}{2}\\ &{} \frac{1}{3}, &{} 1+\frac{2}{3} \end{array}\mid 1\right) =\frac{\pi \Gamma (\frac{1}{3})\Gamma (\frac{5}{3})}{\Gamma (\frac{7}{24}) \Gamma (\frac{23}{24})\Gamma (\frac{13}{24})\Gamma (\frac{29}{24})}. \end{aligned}$$

(3.1)

This is a special case of Whipple’s theorem. We use this identity to motivate our next result.

If we put $n=3$ in Theorem 3.1, then we have ${_3{\widetilde{G}}_3}(t)_p={_3{\mathbb {G}}_3}\left[ \begin{array}{ccc} \frac{1}{4}, &{} \frac{3}{4}, &{} \frac{1}{2}\\ 0, &{} \frac{1}{3}, &{} \frac{2}{3} \end{array}\mid t\right] _p$ and we obtain

Corollary 3.2

Let $p>3$ be a prime and $t\in {\mathbb {F}}_p^\times$. Let $27y^3-27y^2+4t$ be a polynomial over ${\mathbb {F}}_p$. Then the following are true.

• (1)
  $$\begin{aligned} {_3{\widetilde{G}}_3}(1)_p=1+\varphi (-2)=\left\{ \begin{array}{ll} 2, &{} \hbox {if} ~p\equiv 1,3\pmod {8};\\ 0, &{} \hbox {if} ~p\equiv 5,7\pmod {8}. \end{array} \right. \end{aligned}$$

  (3.2)

• (2) Let $t\ne 1$ and $27y^3-27y^2+4t$ be irreducible over ${\mathbb {F}}_p$. Then ${_3{\widetilde{G}}_3}(t)_p=0$.

• (3) Let $t\ne 1$ and $27y^3-27y^2+4t$ has one root $a\in {\mathbb {F}}_p$ counting with multiplicity. Then ${_3{\widetilde{G}}_3}(t)_p=\varphi (a(a-1))$.

• (4) Let $t\ne 1$ and $a_1,a_2, a_3$ be the roots of the polynomial $27y^3-27y^2+4t$ in ${\mathbb {F}}_p$. Then ${_3{\widetilde{G}}_3}(t)_p=\displaystyle \sum _{i=1}^3\varphi (a_i(a_i-1))$.

Equation (3.2) can be viewed as a p-adic analogue of (3.1) which is a special case of Whipple’s theorem. This corollary extends (3.1) over ${\mathbb {F}}_p$ in the p-adic setting.

Remark

It is of interest to discuss the zeros of the function ${_n{\widetilde{G}}_n}(t)_p$ over ${\mathbb {F}}_p$. For example, if $t=1$, then ${_3{\widetilde{G}}_3}(1)_p=0$ if and only if $p\equiv 5,7\pmod {8}$. Therefore, $t=1$ is a non trivial zero of the function ${_3{\widetilde{G}}_3}(t)_p$ for infinitely many primes p. Let $t\in {\mathbb {Q}}-\{0,1\}$ and $27y^3-27y^2+4t$ be irreducible over ${\mathbb {F}}_p$ for infinitely primes p, then ${_3{\widetilde{G}}_3}(t)_p=0$ for infinitely many primes p. However, we do not know whether such t exist or not. This will be an interesting question to study in future.

We now investigate two more general cases of zeros of p-adic hypergeometric functions. By Theorem 3.1, we obtain

Corollary 3.3

Let $n>3$ be an even integer and p be an odd prime satisfying $p\not \mid n(n-1)$. If $t\in {\mathbb {F}}_p$, then ${_n{\widetilde{G}}_n}(t)_p=0$ if and only if $t=0$. In other words, if $t\ne 0$, then ${_n{\widetilde{G}}_n}(t)_p\ne 0$ for all primes $p\not \mid n(n-1)$.

Remark

This corollary is in contrast to Corollary 2.7. If $p\mid n(n-1)$, then the function ${_n{\widetilde{G}}_n}(t)_p$ is not defined for these primes. Hence, by Corollary 3.3, we obtain that if $t\ne 0$, then there is no prime p for which ${_n{\widetilde{G}}_n}(t)_p=0$.

4 Notation and preliminaries

4.1 Multiplicative characters

Let $\widehat{{\mathbb {F}}_p^\times }$ denote the group of all multiplicative characters of ${\mathbb {F}}_p^{\times }$. Let ${\overline{\chi }}$ denote the inverse of a multiplicative character $\chi$. We extend the domain of each $\chi \in \widehat{{\mathbb {F}}_p^\times }$ to ${\mathbb {F}}_p$ by simply setting $\chi (0):=0$ including the trivial character $\varepsilon .$ We start with a lemma that gives an orthogonality relation of multiplicative characters.

Lemma 4.1

[15, Chap. 8] Let p be an odd prime. Then

$$\begin{aligned} \sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}\chi (x)=\left\{ \begin{array}{ll} p-1 , &{} \hbox {if} ~x=1; \\ 0, &{} \hbox {if} ~x\ne 1. \end{array} \right. \end{aligned}$$

(4.1)

Let $\zeta _p$ denote a fixed primitive p-th root of unity. For multiplicative character $\chi$ of ${\mathbb {F}}_p^{\times }$ the Gauss sum is defined by

$$\begin{aligned} g(\chi ):=\sum \limits _{x\in {\mathbb {F}}_p}\chi (x)~\zeta _p^x. \end{aligned}$$

If $\chi =\varepsilon$, then it is easy to see that $g(\varepsilon )=-1$. For more details on the properties of the Gauss sum, see [11]. Let $\delta : \widehat{{\mathbb {F}}_p^\times }\rightarrow \{0,1\}$ be defined by

$$\begin{aligned} \delta (\chi )=\left\{ \begin{array}{ll} 1 , &{} \hbox {if} \chi =\varepsilon ; \\ 0, &{} \hbox {if} \chi \ne \varepsilon . \end{array} \right. \end{aligned}$$

(4.2)

We now state a product formula for Gauss sums.

Lemma 4.2

[13, Eq. 1.12] Let $\chi \in \widehat{{\mathbb {F}}_p^\times }$. Then

$$\begin{aligned} g(\chi )g({\overline{\chi }})=p\cdot \chi (-1)-(p-1)\delta (\chi ). \end{aligned}$$

(4.3)

For multiplicative characters $\chi$ and $\psi$ of ${\mathbb {F}}_p$ the Jacobi sum is defined by

$$\begin{aligned} J(\chi ,\psi ):=\sum _{y\in {\mathbb {F}}_p}\chi (y)\psi (1-y), \end{aligned}$$

(4.4)

and the normalized Jacobi sum known as binomial is defined by

$$\begin{aligned} {\chi \atopwithdelims ()\psi }:=\frac{\psi (-1)}{p}J(\chi ,{\overline{\psi }}). \end{aligned}$$

(4.5)

The following relation provides a relation between Gauss and Jacobi sums.

Lemma 4.3

[13, Eq. 1.14] Let $\chi _1,\chi _2\in \widehat{{\mathbb {F}}_p^\times }$. Then

$$\begin{aligned} J(\chi _1,\chi _2)=\frac{g(\chi _1)g(\chi _2)}{g(\chi _1\chi _2)}+(p-1)\chi _2(-1)\delta (\chi _1\chi _2). \end{aligned}$$

(4.6)

The following product formula of Hasse-Davenport is very important.

Theorem 4.4

[11, Hasse-Davenport relation, Theorem 11.3.5] Let $\psi$ be a multiplicative character of ${\mathbb {F}}_p^\times$ of order m for some positive integer m. For a multiplicative character $\chi$ of ${\mathbb {F}}_p^\times$, we have

$$\begin{aligned} \prod _{i=0}^{m-1}g(\chi \psi ^i)=g(\chi ^m)\chi ^{-m}(m)\prod _{i=1}^{m-1}g(\psi ^i). \end{aligned}$$

(4.7)

We now recall certain properties of binomial from [13, eq. 2.12,eq. 2.7].

$$\begin{aligned} {\chi \atopwithdelims ()\varepsilon }={\chi \atopwithdelims ()\chi }=\frac{-1}{p}+\frac{p-1}{p}\delta (\chi ), \end{aligned}$$

(4.8)

and

$$\begin{aligned} {\chi \atopwithdelims ()\psi }={\chi \atopwithdelims ()\chi {\overline{\psi }}}. \end{aligned}$$

(4.9)

Proposition 4.5

[19, Proposition 2.37] Let p is a prime and $\gcd {(b,p)}=1$. If n is a positive integer and $d=\gcd {(n,p-1)}$, then the congruence $y^n\equiv b\pmod {p}$ has d solutions or no solution according as $b^{\frac{p-1}{d}}\equiv 1\pmod {p}$ or not.

4.2 p-adic preliminaries

Let $\overline{{\mathbb {Q}}_p}$ denote the algebraic closure of ${\mathbb {Q}}_p$ and ${\mathbb {C}}_p$ denote the completion of $\overline{{\mathbb {Q}}_p}$. For a positive integer n, the p-adic gamma function $\Gamma _p(n)$ is defined as follows:

$$\begin{aligned} \Gamma _p(n):=(-1)^n\prod \limits _{0<j<n,p\not \mid j}j. \end{aligned}$$

It can be extended to all $x\in {\mathbb {Z}}_p$ by setting $\Gamma _p(0):=1$ and for $x\ne 0$

$$\begin{aligned} \Gamma _p(x):=\lim _{x_n\rightarrow x}\Gamma _p(x_n), \end{aligned}$$

where $\{x_n\}$ is a sequence of positive integers p-adically approaching to x. Let $\pi \in {\mathbb {C}}_p$ be the fixed root of $x^{p-1} + p=0$ and $\pi \equiv \zeta _p-1 \pmod {(\zeta _p-1)^2}$. The result given below is known as Gross-Koblitz formula.

Theorem 4.6

[14, Gross-Koblitz formula] For $j\in {\mathbb {Z}}$,

$$\begin{aligned} g({\overline{\omega }}^j)=-\pi ^{(p-1)\langle \frac{j}{p-1} \rangle }\Gamma _p\left( \left\langle \frac{j}{p-1} \right\rangle \right) . \end{aligned}$$

An important product formula of p-adic gamma functions is given below. If $m\in {\mathbb {Z}}^+$, $p\not \mid m$, and $x=\frac{r}{p-1}$ with $0\le r\le p-1$ then

$$\begin{aligned} \prod _{h=0}^{m-1}\Gamma _p\left( \frac{x+h}{m}\right) =\omega (m^{(1-x)(1-p)})~\Gamma _p(x)\prod _{h=1}^{m-1}\Gamma _p\left( \frac{h}{m}\right) . \end{aligned}$$

(4.10)

The next two relations are essentially contained in [18]. Let $t\in {\mathbb {Z}}^{+}$ and $p\not \mid t$. Then for $0\le j\le p-2$, we have

$$\begin{aligned} \omega (t^{tj})\Gamma _p\left( \left\langle \frac{tj}{p-1}\right\rangle \right) \prod _{h=1}^{t-1}\Gamma _p\left( \frac{h}{t}\right) =\prod _{h=0}^{t-1}\Gamma _p\left( \left\langle \frac{h}{t}+\frac{j}{p-1}\right\rangle \right) , \end{aligned}$$

(4.11)

and

$$\begin{aligned} \omega (t^{-tj})\Gamma _p\left( \left\langle \frac{-tj}{p-1}\right\rangle \right) \prod _{h=1}^{t-1}\Gamma _p\left( \frac{h}{t}\right) =\prod _{h=1}^{t}\Gamma _p\left( \left\langle \frac{h}{t}-\frac{j}{p-1}\right\rangle \right) . \end{aligned}$$

(4.12)

Lemma 4.7

Let $m\ge 1$ be positive integer and p be an odd prime. For $1\le j\le p-2$ we have $\left\lfloor \frac{mj}{p-1}\right\rfloor =\displaystyle \sum _{h=0}^{m-1}\left\lfloor \frac{h}{m}+\frac{j}{p-1}\right\rfloor$, $\left\lfloor \frac{1}{2}-\frac{mj}{p-1}\right\rfloor =\displaystyle \sum _{h=0}^{m-1}\left\lfloor \frac{1+2h}{2m}-\frac{j}{p-1}\right\rfloor$, and $\left\lfloor \frac{-2j}{p-1}\right\rfloor =-1+\left\lfloor \frac{1}{2}-\frac{j}{p-1}\right\rfloor$.

Proof

As $\left\lfloor \dfrac{mj}{p-1}\right\rfloor =0,1,\ldots ,$ or $m-1$, it gives $\left\lfloor \dfrac{mj}{p-1}\right\rfloor =\displaystyle \sum _{h=0}^{m-1} \left\lfloor \frac{h}{m}+\frac{j}{p-1}\right\rfloor$. Similarly, it is straightforward to verify the remaining equalities. $\square$

5 Proof of main theorems

Before going to prove the main results regarding the values of ${_nG_n}(t)$ for arbitrary t, we prove two propositions.

Proposition 5.1

Let $n\ge 3$ be an integer and p be an odd prime such that $p\not \mid n$. For $t\in {\mathbb {F}}_p^{\times }$, let $B_n(t)=\displaystyle \sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}g(\varphi \chi ^n)g({\overline{\chi }}^n){\overline{\chi }}((-1)^nt)$. If t is n-th power residue modulo p, then we have

$$\begin{aligned} B_n(t)=(p-1)g(\varphi )\displaystyle \sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ a^n\equiv t\pmod {p} \end{array}}\varphi (a)\varphi (a-1). \end{aligned}$$

Otherwise, $B_n(t)=0$.

Proof

By (4.6), and (4.9), it follows that

$$\begin{aligned} B_n(t)&=\sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}g(\varphi \chi ^n)g({\overline{\chi }}^n){\overline{\chi }}((-1)^nt) =pg(\varphi )\sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}{\varphi \chi ^n\atopwithdelims ()\varphi }{\overline{\chi }}(t)\nonumber \\&=\varphi (-1)g(\varphi )\sum _{y\in {\mathbb {F}}_p}\varphi (y)\varphi (1-y)\sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}\chi \left( \frac{y^n}{t}\right) . \end{aligned}$$

(5.1)

(4.1) gives that the latter sum present in (5.1) is non zero only if $y^n\equiv t\pmod {p}$ has a solution in ${\mathbb {F}}_p^{\times }$. Therefore, if t is n-th power residue modulo p, then we obtain

$$\begin{aligned} B_n(t)=(p-1)g(\varphi )\displaystyle \sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ a^n\equiv t\pmod {p} \end{array}}\varphi (a)\varphi (a-1). \end{aligned}$$

On the other hand, if t is not a n-th power residue modulo p, then we have $B_{n}(t)=0$. This completes the proof of the proposition. $\square$

Proposition 5.2

Let $n\ge 3$ be an integer and $p\ge 3$ be a prime such that $p\not \mid n$. For $t\in {\mathbb {F}}_p^{\times }$, let $B_n(t)=\displaystyle \sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}g(\varphi \chi ^n)g({\overline{\chi }}^n){\overline{\chi }}((-1)^nt)$. Then

$$\begin{aligned} B_n(t)=(p-1)g(\varphi )~ {_nG_n}(t)_p. \end{aligned}$$

Proof

Taking $\chi =\omega ^j$ and applying Gross-Koblitz formula, we have

$$\begin{aligned} B_n(t)&=\sum _{j=0}^{p-2}\pi ^{(p-1)\ell _j}~\Gamma _p\left( \left\langle \frac{1}{2}-\frac{nj}{p-1}\right\rangle \right) \Gamma _p\left( \left\langle \frac{nj}{p-1}\right\rangle \right) {\overline{\omega }}^j((-1)^nt), \end{aligned}$$

(5.2)

where $\ell _j=\frac{1}{2} -\left\lfloor \frac{1}{2}-\frac{nj}{p-1}\right\rfloor -\left\lfloor \frac{nj}{p-1}\right\rfloor$. Applying (4.10) (with $x=\left\langle \frac{1}{2}-\frac{nj}{p-1}\right\rangle$, and $m=n$) we obtain

$$\begin{aligned} \prod _{h=0}^{n-1}\Gamma _p\left( \left\langle \frac{1+2h}{2n}-\frac{j}{p-1}\right\rangle \right) =\frac{{\overline{\omega }}^{nj}(n)}{\varphi (n)}\cdot \Gamma _p\left( \left\langle \frac{1}{2}-\frac{nj}{p-1}\right\rangle \right) \prod _{h=1}^{n-1}\Gamma _p\left( \frac{h}{n}\right) . \end{aligned}$$

(5.3)

Also, (4.11) yields

$$\begin{aligned} \omega ^{nj}(n)\Gamma _p\left( \left\langle \frac{nj}{p-1}\right\rangle \right) \prod _{h=1}^{n-1}\Gamma _p\left( \frac{h}{n}\right) =\prod _{h=0}^{n-1}\Gamma _p\left( \left\langle \frac{h}{n}+\frac{j}{p-1}\right\rangle \right) . \end{aligned}$$

(5.4)

By Lemma 4.7, we have

$$\begin{aligned} \ell _j=\frac{1}{2}-\sum _{h=0}^{n-1}\left\{ \left\lfloor \frac{1+2h}{2n}-\frac{j}{p-1}\right\rfloor + \left\lfloor \frac{h}{n}+\frac{j}{p-1}\right\rfloor \right\} \end{aligned}$$

(5.5)

Putting (5.3), (5.4), and (5.5) in (5.2), we obtain

$$\begin{aligned} \frac{B_n(t)}{1-p}=\pi ^{\frac{(p-1)}{2}}\varphi (n)\prod _{h=0}^{n-1}\frac{\Gamma _p(\frac{1+2h}{2n})}{\Gamma _p(\frac{h}{n})}~{_n{G}_n}(t)_p. \end{aligned}$$

(5.6)

Using (5.3) with $j=0$, we obtain $\displaystyle \prod _{h=0}^{n-1}\frac{\Gamma _p(\frac{1+2h}{2n})}{\Gamma _p(\frac{h}{n})}=\varphi (n)\Gamma _p\left( \frac{1}{2}\right)$. Using this, and the Gross-Koblitz formula in (5.6), we obtain

$$\begin{aligned} B_n(t)=(p-1)g(\varphi )~ {_nG_n}(t)_p. \end{aligned}$$

This completes the proof. $\square$

Proof of Theorem 2.1

Let $B_n(t)=\displaystyle \sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}g(\varphi \chi ^n)g({\overline{\chi }}^n){\overline{\chi }}((-1)^nt)$. Applying Proposition 5.2 on the above sum, we obtain

$$\begin{aligned} B_n(t)=(p-1)g(\varphi )~ {_nG_n}(t)_p. \end{aligned}$$

(5.7)

Let $t^\frac{(p-1)}{d}\equiv 1\pmod {p}$. Then applying Proposition 4.5 (with $b=t$), we obtain that t is n-th power residue modulo p. Using this information and Proposition 5.1, we obtain that

$$\begin{aligned} B_n(t)=(p-1)g(\varphi )\sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ a^n\equiv t\pmod {p} \end{array}}\varphi (a)\varphi (a-1). \end{aligned}$$

(5.8)

Combining (5.7) and (5.8), we complete the proof of the first part. Similarly, we prove the second part. $\square$

Proof of Theorem 2.2

If $p\equiv 1\pmod {3}$, then we have $\gcd {(3,p-1)}=3$. For $t\in {\mathbb {F}}_p\setminus \{0,1\}$, let us assume that ${_3G_3}(t)=0$. Since g is a primitive root modulo p, so we write $t=g^i$ for some integer i. If $\gcd {(i,3)}=3$, then $i=3k$ for some integer k. This yields

$$\begin{aligned} t^{\frac{p-1}{3}}=g^{\frac{3k(p-1)}{3}}\equiv 1\pmod {p}. \end{aligned}$$

If we use Theorem 2.1 for $n=3$, then we obtain

$$\begin{aligned} {_3G_3}(t)_p=\sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ a^3\equiv t\pmod {p} \end{array}}\varphi (a(a-1)). \end{aligned}$$

(5.9)

Since $y^3\equiv t\pmod {p}$ has 3 roots as $p\equiv 1\pmod {3}$, so the R.H.S. of (5.9) cannot be equal to zero. However, this is a contradiction to the fact that ${_3G_3}(t)_p=0$. Therefore, we have $\gcd {(i,3)}=1$. Conversely, suppose that $\gcd {(i,3)}=1$. Then

$$\begin{aligned} t^{\frac{p-1}{3}}=g^{\frac{i(p-1)}{3}}\not \equiv 1\pmod {p}. \end{aligned}$$

Again, if we use Theorem 2.1 for $n=3$, then we have ${_3G_3}(t)_p=0$. Now, if $t=1$ then $y^3\equiv 1\pmod {p}$ has three roots in ${\mathbb {F}}_p$. Let $a\ne 1$ be a solution of $y^3\equiv 1\pmod {p}$ in ${\mathbb {F}}_p^{\times }$. Then the complete list of solutions of this congruence are 1, a, $a^2=a^{-1}$, where $a^{-1}$ denotes the inverse of a in ${\mathbb {F}}_p^\times$. Using this information and Theorem 2.1 (with $n=3$, and $t=1$), we have

$$\begin{aligned} {_3G_3}(1)_p=\varphi (a)\varphi (a-1)+\varphi (a^{-1})\varphi (a^{-1}-1). \end{aligned}$$

(5.10)

Now, ${_3G_3}(1)_p=0$ if and only if $\varphi (a)\varphi (a-1)+\varphi (a^{-1})\varphi (a^{-1}-1)=0$. This is possible if and only if $1+\varphi (-a)=0$. This is equivalent to

$$1+\varphi (-1)\varphi (a^{-1})=1+\varphi (-1)\varphi (a^{2})=1+\varphi (-1)=0.$$

This is true if and only if $p\equiv 3\pmod {4}$. As $p\equiv 1\pmod {3}$ it follows that ${_3G_3}(1)_p=0$ if and only if $p\equiv 7\pmod {12}$. To prove the second part of the theorem, let $p\not \equiv 1\pmod {3}$. Then $\gcd {(3,p-1)}=1$, and for $t\in {\mathbb {F}}_p^{\times }$, it is well known that $t^{p-1}\equiv 1\pmod {p}$. For $t\ne 0,1$ applying Theorem 2.1 (with $n=3$) and Proposition 4.5, we obtain

$$\begin{aligned} {_3G_3}(t)_p=\varphi (a(a-1)), \end{aligned}$$

(5.11)

where $a^3\equiv t\pmod {p}$. The R.H.S. of (5.11) can take only values $\pm 1$. Thus, if $t\ne 0,1$ then ${_3G_3}(t)\ne 0$. Now, if $t=1$, then the only solution of $y^3\equiv 1\pmod {p}$ is 1. Using this information and Theorem 2.1, we obtain that ${_3G_3}(1)_p=0$. This completes the proof. $\square$

Proof of Corollary 2.3

From Theorem 2.2, we obtain that ${_3G_3}(1)_p=0$ if $p\equiv 5,7,11\pmod {12}$. Now, if $p\equiv 1\pmod {12}$, then the congruence $y^3\equiv 1\pmod {p}$ has three solutions that are given by 1, $\lambda$ and $\lambda ^2$ such that $1+\lambda +\lambda ^2\equiv 0\pmod {p}.$ If we use Theorem 2.1 (for $n=3$), then we can write

$$\begin{aligned} {_3G_3}(1)_p=\varphi (\lambda )\varphi (\lambda -1)+\varphi (\lambda ^2)\varphi (\lambda ^2-1). \end{aligned}$$

Note that modulo p we have $\lambda \equiv \lambda ^{-2},$ $1+\lambda \equiv -\lambda ^2$ and $\varphi (-1)=1.$ Using these, we have

$$\begin{aligned} {_3G_3}(1)_p=2\cdot \varphi (\lambda -1). \end{aligned}$$

Moreover, we have $(\lambda -1)^2\equiv -3\lambda \pmod {p}.$ Therefore, finally using this we complete the proof. $\square$

Proof of Corollary 2.4

The first part of the corollary follows easily from Theorem 2.1. Now, if $p\equiv 2\pmod {3}$, then $\gcd {(3,p-1)=1}$ and $t^{p-1}\equiv 1\pmod {p}$ for all t such that $\gcd {(t,p)}=1$. Using this and Proposition 4.5, we obtain that the congruence $y^3\equiv t\pmod {p}$ has a unique solution. To be specific, it can be easily shown that the congruence has the unique solution $y\equiv t^{\frac{2p-1}{3}}\pmod {p}$. Using this fact and Theorem 2.1, we conclude the result. $\square$

Proof of Corollary 2.5

By Theorem 2.1, we have

$$\begin{aligned} {_nG_n}(t)_p=\sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ a^n\equiv t\pmod {p} \end{array}}\varphi (a)\varphi (a-1). \end{aligned}$$

(5.12)

Now, if n is even, then d is even. Let $t\ne 1$. By Proposition 4.5, we know that the congruence $y^n\equiv t\pmod {p}$ has d solutions. If $a_1,a_2,\ldots ,a_d$ are the modulo p solutions of the congruence $y^n\equiv t\pmod {p}$ such that $\displaystyle \sum _{i=1}^{d}\varphi (a_i(a_i-1))=0$, then (5.12) gives that ${_nG_n}(t)_p=0$. Let $t=1$ and $a_1,a_2,\ldots ,a_{d-1}$ are the solutions of $y^n\equiv 1\pmod {p}$ modulo p different from 1. If possible let ${_nG_n}(1)_p=0$. Then (5.12) gives $\displaystyle \sum _{i=1}^{d-1}\varphi (a_i(a_i-1))=0$, which is not possible as d is even, so ${_nG_n}(1)_p\ne 0$. To prove the second part let n be odd. Then d is odd. If $t\ne 1$, then using Proposition 4.5, we obtain that the congruence $y^n\equiv t\pmod {p}$ has d solutions. If $a_1,a_2,\ldots ,a_d$ are the solutions of the congruence $y^n\equiv t\pmod {p}$, then (5.12) gives ${_nG_n}(t)_p=\displaystyle \sum _{i=1}^{d}\varphi (a_i(a_i-1)),$ which cannot be equal to zero as d is odd. Similarly, we settle the case for $t=1$. $\square$

Proof of Corollary 2.6

If $\gcd {(n,p(p-1))}=1$, then it follows from Proposition 4.5 that the congruence

$$y^n\equiv t\pmod {p}$$

has a unique solution for each t such that $\gcd {(t,p)}=1$. Now, if $t=1$, then 1 is the unique solution of the congruence $y^n\equiv 1\pmod {p}$. Using this information in Theorem 2.1 we have ${_nG_n}(1)_p=0$. Let $t\ne 1$ and $y\equiv a\pmod {p}$ be the unique solution of $y^n\equiv t\pmod {p}$. Then Theorem 2.1 yields ${_nG_n}(t)_p=\varphi (a(a-1))$, which cannot be zero. This completes the proof. $\square$

Proof of Corollary 2.7

If n is even and $p\equiv 3\pmod {4}$ then $\gcd {(n,p-1)}=2k$ for some odd integer k. This gives $(-1)^{\frac{p-1}{2k}}\equiv -1\pmod {p}$. Then it follows from Theorem 2.1 that ${_nG_n}(-1)_p=0$. This completes the proof. $\square$

We now provide two propositions. These propositions are used to examine the values of the function ${_n{\widetilde{G}}_n}(t)$.

Proposition 5.3

Let $n\ge 3$ be an integer and p be a prime such that $p\not \mid n(n-1)$. Let $t\in {\mathbb {F}}_p^{\times }$ and $f_t(y)=y^{n}-y^{n-1}+\frac{(n-1)^{n-1}}{n^n}t\in {\mathbb {F}}_p[y]$ be a polynomial in y. Let $\alpha =\frac{4(1-n)^{n-1}}{n^n}$, and $\beta _n(t)=\left\{ \begin{array}{ll} 1, &{} \hbox {if} ~n ~is odd;\\ 1-(p-1)\varphi ((1-n) t), &{} \hbox {if} ~n ~\hbox {is even.} \end{array} \right.$

Let $A_n(t)=\displaystyle \sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}g(\varphi \chi ^{n-1})g({\overline{\chi }}^n)g({\overline{\chi }}) g(\chi ^2){\overline{\chi }}(\alpha t)$. Then

$$\begin{aligned} A_n(t)=p(p-1)g(\varphi )\displaystyle \sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ f_t(a)\equiv 0\pmod {p} \end{array}}\varphi (a(a-1))+(p-1)g(\varphi )\beta _n(t). \end{aligned}$$

Proof

Multiplying both numerator and denominator by $g(\varphi {\overline{\chi }})$, we have

$$\begin{aligned} A_t&=\sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}\frac{g(\varphi \chi ^{n-1})g({\overline{\chi }}^n)}{g(\varphi {\overline{\chi }})} g(\varphi {\overline{\chi }})g({\overline{\chi }})g(\chi ^2){\overline{\chi }}(\alpha t). \end{aligned}$$

(5.13)

Applying (4.7) (with $m=2$), we have $g(\varphi {\overline{\chi }})g({\overline{\chi }})=g({\overline{\chi }}^2)g(\varphi )\chi (4)$. Substituting this in (5.13) and then using (4.6), (4.3), and (4.8), we obtain

$$\begin{aligned} A_n(t)&=pg(\varphi )\sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}J(\varphi \chi ^{n-1},{\overline{\chi }}^n) {\overline{\chi }}\left( \frac{\alpha t}{4}\right) +(p-1)g(\varphi )\beta _n(t). \end{aligned}$$

By (4.5) and (4.9), we re-write $A_n(t)$ as follows:

$$\begin{aligned} A_n(t)&=p^2g(\varphi )\sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}{\varphi \chi ^{n-1}\atopwithdelims ()\varphi {\overline{\chi }}}{\overline{\chi }} \left( \frac{(-1)^{n}\alpha t}{4}\right) +(p-1)g(\varphi )\beta _n(t)\nonumber \\&=pg(\varphi )\sum _{1\ne y\in {\mathbb {F}}_p}\varphi \left( \frac{y}{y-1}\right) \sum _{\chi \in \widehat{{\mathbb {F}}_p}^\times }\chi \left( \frac{4y^{n-1}(1-y)}{(-1)^{n-1}\alpha t}\right) +(p-1)g(\varphi ) \beta _n(t). \end{aligned}$$

(5.14)

By (4.1), we obtain that the second sum present on the R.H.S. of (5.14) is non zero only if $y^n-y^{n-1}+\frac{(n-1)^{n-1}}{n^n}t\equiv 0\pmod {p}$ admits a solution in ${\mathbb {F}}_p$. Using this information, we have

$$\begin{aligned} A_n(t)=p(p-1)g(\varphi )\displaystyle \sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ f_t(a)\equiv 0\pmod {p} \end{array}} \varphi (a(a-1))+(p-1)g(\varphi )\beta _n(t) \end{aligned}$$

$\square$

Proposition 5.4

Let $n\ge 3$ and $p\not \mid n(n-1)$ be an odd prime. If $t\in {\mathbb {F}}_p^{\times }$ and $\alpha =\frac{4(1-n)^{n-1}}{n^n}$, then let $A_n(t)=\displaystyle \sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}g(\varphi \chi ^{n-1})g({\overline{\chi }}^n)g({\overline{\chi }}) g(\chi ^2){\overline{\chi }}\left( \alpha t\right) .$ Then we have that

$$\begin{aligned} A_n(t)=(p-1)g(\varphi )(1+p\cdot {_n{\widetilde{G}}_n}(t)_p). \end{aligned}$$

Proof

Replacing $\chi$ by $\omega ^j$ and then applying Gross-Koblitz formula, (4.10), (4.11), and (4.12) similarly as shown in the proof of Proposition 5.2, we deduce that

$$\begin{aligned} \frac{A_n(t)}{\varphi (n-1)}&=\sum _{j=1}^{p-2}\pi ^{(p-1)\ell _j}~{\overline{\omega }}^j((-1)^{n-1}t) \prod _{h=0}^{n-2}\frac{\Gamma _p\left( \left\langle \frac{1+2h}{2(n-1)}-\frac{j}{p-1}\right\rangle \right) }{\Gamma _p(\frac{h}{n-1})}\nonumber \\&\times \frac{\Gamma _p\left( \left\langle \frac{1}{2}-\frac{j}{p-1}\right\rangle \right) }{\Gamma _p(\frac{1}{2})} \prod _{h=0}^{n-1}\frac{\Gamma _p\left( \left\langle \frac{h}{n}+\frac{j}{p-1}\right\rangle \right) }{\Gamma _p(\frac{h}{n})} \nonumber \\&\times \Gamma _p\left( \left\langle \frac{j}{p-1}\right\rangle \right) \Gamma _p\left( \left\langle 1-\frac{j}{p-1}\right\rangle \right) +\pi ^{\frac{p-1}{2}}\prod _{h=0}^{n-2} \frac{\Gamma _p\left( \frac{1+2h}{2(n-1)}\right) }{\Gamma _p(\frac{h}{n-1})}, \end{aligned}$$

(5.15)

where $\ell _j=\frac{1}{2}-\lfloor \frac{1}{2}-\frac{(n-1)j}{p-1}\rfloor -\lfloor \frac{nj}{p-1}\rfloor -\lfloor \frac{j}{p-1}\rfloor -\lfloor \frac{-2j}{p-1}\rfloor$. If $1\le j\le p-2$ then the Gross-Koblitz formula, and (4.3) give

$$\begin{aligned} \Gamma _p\left( \left\langle \frac{j}{p-1}\right\rangle \right) \Gamma _p\left( \left\langle 1-\frac{j}{p-1}\right\rangle \right) =-\omega ^j(-1). \end{aligned}$$

(5.16)

Also, Gross-Koblitz gives

$$\begin{aligned} g(\varphi )=-\pi ^{(p-1)/2}~\Gamma _p(1/2). \end{aligned}$$

(5.17)

(4.10) yields $\displaystyle \prod _{h=0}^{n-2}\frac{\Gamma _p(\frac{1+2h}{2(n-1)})}{\Gamma _p(\frac{h}{n-1})}=\varphi (n-1) \Gamma _p(1/2)$. Substituting this identity along with (5.16), and (5.17) into (5.15) and finally using Lemma 4.7 in the expression of $\ell _j$, we deduce the required identity. $\square$

Proof of Theorem 3.1

Let $A_n(t)=\displaystyle \sum _{\chi \in \widehat{{\mathbb {F}}_p^\times }}g(\varphi \chi ^{n-1})g({\overline{\chi }}^n)g({\overline{\chi }}) g(\chi ^2){\overline{\chi }}\left( \alpha t\right) ,$ where $\alpha =\frac{4(1-n)^{n-1}}{n^n}$. Now, applying Proposition 5.3, and Proposition 5.4 on $A_n(t)$ and then combining both the expressions we obtain the result. $\square$

Proof of Corollary 3.2

Let $n=3$ and $f_t(y)=27y^3-27y^2+4t$. Then applying Theorem 3.1 for $n=3$ and $p>3$ we obtain

$$\begin{aligned} {_3{\widetilde{G}}_3}(t)_p=\sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ f_t(a)\equiv 0\pmod {p} \end{array}}\varphi (a(a-1)). \end{aligned}$$

(5.18)

For $p>3$, we know that if $t=1$, then the roots of the polynomial $27y^3-27y^2+4$ are $\frac{2}{3}$ with multiplicity two, and $\frac{-1}{3}$ with one. Therefore, if $t=1$, then using this information in (5.18), we obtain

$$\begin{aligned} {_3{\widetilde{G}}_3}(1)_p=1+\varphi (-2). \end{aligned}$$

This proves the first part. Similarly, we prove the other parts of the corollary. $\square$

Proof of Corollary 3.3

If n is even and $t\in {\mathbb {F}}_p^\times$, then Theorem 3.1 yields

$$\begin{aligned} {_n{\widetilde{G}}_n}(t)_p=\frac{(1-p)\varphi ((1-n)t)}{p}+\sum _{\begin{array}{c} a\in {\mathbb {F}}_p\\ f_t(a)\equiv 0\pmod {p} \end{array}} \varphi (a(a-1)). \end{aligned}$$

(5.19)

If possible let ${_n{\widetilde{G}}_n}(t)_p=0$, then (5.19) gives $(p-1)\varphi (t)\varphi (1-n)\equiv 0\pmod {p}$, which is not possible. Hence, if $t\ne 0$, then ${_n{\widetilde{G}}_n}(t)_p\ne 0$. This completes the proof. $\square$