Diophantine equations with Appell sequences

Abstract

We consider the Diophantine equation $P_n (x) = g(y)$ in $x, y$ where $P_n (x), g(x) \in \mathbb {Q}[x], \deg g(x) \ge 3$ and $\left\{ P_n (x)\right\} _{n \ge 0}$ is an Appell sequence. Under some reasonable assumptions on $P_n(x)$ we prove an ineffective finiteness result on the above equation.

1 Introduction

For $n \in \mathbb {N} \cup \left\{ 0\right\}$, let $P_n (x)$ be a polynomial with rational coefficients and with $\deg P_n (x) = n$. Further, let $P_0 (x)$ be a non-zero constant. The sequence $\left\{ P_n (x)\right\} _{n \ge 0}$ is called an Appell sequence (and $P_n (x)$ is called an Appell polynomial) if

$$\begin{aligned} P_n^{\prime } (x) = n P_{n-1} (x) \quad \text { for all } \quad n \in \mathbb {N}. \end{aligned}$$

(1.1)

The history of such polynomials goes back to Appell’s work [2] in 1880. There are several well-known examples of Appell sequences, such as the Bernoulli polynomials $B_n (x)$, the Euler polynomials $E_n (x)$, and the Hermite polynomials $H_n(x)$, respectively defined by the following generating series (see e.g. [12])

$$\begin{aligned} \frac{t\exp (tx)}{\exp (t)-1}&= \sum _{n=0}^{\infty }B_{n}(x)\frac{t^n}{n!};\\ \frac{2 \exp (xt)}{\exp (t) + 1}&= \sum ^{\infty }_{n=0}{E_n (x) \frac{t^n}{n!}} \quad (|t| < \pi ); \\ \frac{\exp (tx)}{\exp (t^2 /2)}&= \sum _{n=0}^{\infty }H_{n}(x)\frac{t^n}{n!}. \end{aligned}$$

The above defined Hermite polynomials $H_n(x)$ are sometimes denoted by $He_n(x)$, e.g. in Abramowitz and Stegun [1].

The following properties of Appell polynomials will often be used in the text, sometimes without special reference.

We recall the so-called Appell Identity:

$$\begin{aligned} P_n (x+y) = \sum _{k=0}^n{\left( {\begin{array}{c}n\\ k\end{array}}\right) P_k (x) y^{n-k}} = \sum _{k=0}^n{\left( {\begin{array}{c}n\\ k\end{array}}\right) P_k (y) x^{n-k}}, \end{aligned}$$

(1.2)

which, by setting $y=0$, implies that there exists a sequence of rational numbers $\left\{ c_n\right\} _{n \ge 0}$ with $c_0 \ne 0$ such that

$$\begin{aligned} P_n(x) = \sum _{k=0}^n{\left( {\begin{array}{c}n\\ k\end{array}}\right) c_k x^{n-k}}, \quad \text {where }c_k := P_k(0) \ (k \ge 0). \end{aligned}$$

(1.3)

For the proofs of (1.2) and (1.3) see, for instance Roman [12].

Let $\mathbb {K}$ be an arbitrary field. We denote by $\mathbb {K}[x]$ the ring of polynomials in the variable $x$ with coefficients from $\mathbb {K}$. A decomposition of a polynomial $F(x)$ over $\mathbb {K}$ is an equality of the following form

$$\begin{aligned} F(x) = G_1 (G_2 (x)) \quad (G_1 (x), G_2 (x) \in \mathbb {K}[x]), \end{aligned}$$

which is nontrivial if

$$\begin{aligned} \deg G_1 (x) > 1 \quad \text {and} \quad \deg G_2 (x) > 1. \end{aligned}$$

Two decompositions $F(x) = G_1 (G_2 (x))$ and $F(x) = H_1 (H_2 (x))$ are said to be equivalent if there exists a linear polynomial $\ell (x) \in \mathbb {K}[x]$ such that $G_1 (x) = H_1 (\ell (x))$ and $H_2 (x) = \ell (G_2 (x))$. The polynomial $F(x)$ is called decomposable over $\mathbb {K}$ if it has at least one nontrivial decomposition over $\mathbb {K}$; otherwise it is said to be indecomposable.

The decomposition of Bernoulli polynomials has been described by Bilu et al. in [6]. Decomposition properties of Euler polynomials were recently investigated by Rakaczki and Kreso [11]. These results can both be summarized as follows: the corresponding polynomial ($B_n(x)$ or $E_n(x)$) is indecomposable over $\mathbb {C}$ for all odd $n$, while, if $n$ is even, then any nontrivial decomposition of the polynomial under consideration over $\mathbb {C}$ is equivalent to one of the form

$$\begin{aligned} \widehat{P}_{n/2} \left( \left( x - \frac{1}{2}\right) ^2\right) , \end{aligned}$$

where $\widehat{P}_{n/2}(x)$ is a polynomial of degree $n/2$ which is indecomposable for every $n$. These results from [6] and [11] suggest the following notion. We say that an Appell sequence $\left\{ P_n (x)\right\} _{n \ge 0}$ is of special type if $P_n (x)$ is indecomposable over $\mathbb {C}$ for all odd $n$, and, for even $n$, every nontrivial decomposition of $P_n (x)$ is equivalent to a decomposition of the form

$$\begin{aligned} P_n(x) = \widehat{P}_{n/2} \left( \left( x - \frac{1}{2}\right) ^2\right) , \end{aligned}$$

(1.4)

with an indecomposable polynomial $\widehat{P}_{n/2}(x)$ over $\mathbb {C}$ of degree $n/2$. Clearly, the polynomials $\left\{ B_n (x)\right\} _{n \ge 0}$ and $\left\{ E_n (x)\right\} _{n \ge 0}$ are of special type.

The theory of polynomial decomposition is strongly connected to the theory of separable Diophantine equations since, in 2000, Bilu and Tichy [5] established their general ineffective finiteness criterion on equations of the form $f(x) = g(y)$. (See Proposition 2.1 below.)

In this paper we study the Diophantine equation

$$\begin{aligned} P_n (x) = g(y) \quad \text {in integers} \, x,y, \end{aligned}$$

(1.5)

where $P_n(x)$ is from an Appell sequence of special type and $g(x) \in \mathbb {Q}[x]$, $\deg g(x) \ge 3$. For technical reasons, we restrict ourselves to Appell sequences $\left\{ P_n (x)\right\} _{n \ge 0}$ for which

$$\begin{aligned} \frac{3 P_2 (-c_1 / c_0)^2 - 2 c_0 P_4 (-c_1 / c_0)}{3 P_2 (-c_1 / c_0)^2 - c_0 P_4 (-c_1 / c_0)} \quad \text { is not a positive integer.} \end{aligned}$$

(1.6)

Remark

In the following table, we give the value of the constant from (1.6) for the case when $P_n(x)$ is a Bernoulli, Euler or an Hermite polynomial, respectively.

$$\begin{aligned} \begin{array}{c|c|c} B_n(x) &{} E_n(x) &{} H_n(x) \\ \hline 9/2 &{} 7/2 &{} \mathrm{undefined} \end{array} \end{aligned}$$

For $P_n (x) = B_n (x)$, Rakaczki [10], and independently Kulkarni and Sury [9] characterized those pairs $(n,g(y))$ for which equation (1.5) has infinitely many integer solutions. Recently, Rakaczki and Kreso [11] proved an analogous result for the case when $P_n (x) = (E_n (0) \pm E_n (x))/2$ (which is not an Appell polynomial anymore). For further related results we refer to [7, 8].

We prove the following.

Theorem 1.1

Let $g(x) \in \mathbb {Q}[x]$ with $\deg g(x) \ge 3$, and suppose that $\left\{ P_n (x)\right\} _{n \ge 0}$ is an Appell sequence of special type with property (1.6). Then for $n \ge 7$, equation (1.5) has only finitely many integer solutions $x,y$, apart from the following cases:

• (i) $g(x) = P_n (h(x))$, where $h(x)$ is a polynomial over $\mathbb {Q}$.

• (ii) $g(x) = \gamma (\delta (x)^m)$, where $m$ is a positive integer.

• (iii) $n$ is even and $g(x) = \widehat{P}_{n/2}(q(x)^2 )$

• (iv) $n$ is even and $g(x) = \widehat{P}_{n/2}( \delta (x) q(x)^2 )$

• (v) $n$ is even and $g(x) = \widehat{P}_{n/2}(c \delta (x)^t)$, where $t \ge 3$ is an odd integer

• (vi) $n$ is even and $g(x) = \widehat{P}_{n/2}( (a \delta (x)^2 + b) q(x)^2)$

Here $a,b,c \in \mathbb {Q}\setminus \left\{ 0\right\}$, $\gamma (x),\delta (x) \in \mathbb {Q}[x]$ are linear polynomials and $q(x) \in \mathbb {Q}[x]$ is a non-zero polynomial.

We prove the above theorem by applying among other things the general finiteness criterion of Bilu and Tichy [5] for equation (1.5). Hence our finiteness result is ineffective.

Remark

For $n \ge 7$, our main result is a common generalization of the aforementioned results of Rakaczki [10], Kulkarni and Sury [9] and Rakaczki and Kreso [11]. In the special cases $P_n(x) \in \left\{ B_n(x), E_n(x)\right\}$, one can exclude the exceptional case (ii) by making use of some specific properties of the Bernoulli or Euler polynomials, respectively. (See [9–11])

2 Auxiliary results

Before proving Theorem 1.1, we collect the results that will be applied in the proof. First, we recall the finiteness criterion of Bilu and Tichy [5]. To do this, we need to define five kinds of so-called standard pairs of polynomials.

Let $\alpha , \beta$ be nonzero rational numbers, $\mu , \nu ,q >0$ and $r \ge 0$ be integers, and let $v(x) \in \mathbb {Q} [x]$ be a nonzero polynomial (which may be constant). Denote by $D_{\mu } (x,\delta )$ the $\mu$-th Dickson polynomial, defined by the functional equation $D_{\mu } (z+\delta /z,\delta ) = z^{\mu } + (\delta /z)^{\mu }$ or by the explicit formula

$$\begin{aligned} D_{\mu } (x,\delta ) = \sum _{i=0}^{\left\lfloor \mu / 2 \right\rfloor }{d_{\mu ,i} x^{\mu - 2i}} \qquad \text {with }\,\, d_{\mu ,i} = \frac{\mu }{\mu - i} \left( {\begin{array}{c}\mu - i\\ i\end{array}}\right) (- \delta )^i. \end{aligned}$$

Two polynomials $f_1(x)$ and $g_1(x)$ are said to form a standard pair over $\mathbb {Q}$ if one of the ordered pairs $(f_1(x),g_1(x))$ or $(g_1(x),f_1(x))$ belongs to the list below. The five kinds of standard pairs are then listed in the following table.

Kind	Explicit form of $\left\{ f_1(x),g_1(x)\right\}$	Parameter restrictions
First	$(x^q, \alpha x^{r} v(x)^q)$	$0 \le r < q, (r,q)=1,\, r + \deg v(x) > 0$
Second	$(x^2, (\alpha x^2 + \beta ) v (x)^2)$	–
Third	$(D_{\mu } (x,\alpha ^{\nu }), D_{\nu } (x,\alpha ^{\mu }))$	$(\mu ,\nu ) = 1$
Fourth	$(\alpha ^{\frac{-\mu }{2}} D_{\mu } (x,\alpha ), -\beta ^{\frac{-\nu }{2}} D_{\nu } (x,\beta ))$	$(\mu ,\nu ) = 2$
Fifth	$((\alpha x^2 - 1)^3, 3x^4 - 4x^3)$	–

The following proposition is a special case of the main result of [5].

Proposition 2.1

Let $f(x),g(x) \in \mathbb {Q} [x]$ be nonconstant polynomials such that the equation $f(x) = g(y)$ has infinitely many solutions in rational integers $x,y$. Then $f = \varphi \circ f_1 \circ \lambda$ and $g = \varphi \circ g_1 \circ \mu$, where $\lambda (x), \mu (x) \in \mathbb {Q} [x]$ are linear polynomials, $\varphi (x) \in \mathbb {Q} [x]$, and $(f_1(x),g_1(x))$ is a standard pair over $\mathbb {Q}$.

For $P(x) \in \mathbb {C}[x]$, a complex number $c$ is said to be an extremum if $P(x) - c$ has multiple roots. The $P$-type of $c$ is defined to be the tuple $(\alpha _1, \ldots , \alpha _s)$ of the multiplicities of the distinct roots of $P(x) - c$ in an increasing order. Obviously, $s < \deg P(x)$ and $\alpha _1 + \ldots + \alpha _s = \deg P(x)$.

Proposition 2.2

For $a \ne 0$ and $k \ge 3, D_{\mu }(x, \alpha )$ has exactly two extrema $\pm 2 \alpha ^{\frac{\mu }{2}}$. If $\mu$ is odd, then both are of $P$-type $(1, 2, 2, \ldots , 2)$. If $\mu$ is even, then $2 \alpha ^{\frac{\mu }{2}}$ is of $P$-type $(1, 1, 2, \ldots , 2)$ and $- 2 \alpha ^{\frac{\mu }{2}}$ is of $P$-type $(2, 2, \ldots , 2)$.

Proof

See, for instance [4, Proposition 3.3]. $\square$

We end this section with two technical results. Let $d_1,e_1 \in \mathbb {Q}^*$ and $d_0,e_0 \in \mathbb {Q}$

Proposition 2.3

Suppose that $\left\{ P_n (x)\right\} _{n \ge 0}$ is an Appell sequence of special type. Then the polynomial $P_n (d_1 x + d_0)$ is not of the form $e_1 x^q + e_0$, with $q \ge 7$.

Proof

We assume the contrary, i.e., that we have

$$\begin{aligned} P_n (d_1 x + d_0) = e_1 x^q + e_0 \end{aligned}$$

(2.1)

with $q \ge 7$. Obviously, we then have $n=q$.

We observe from (1.2) and (2.1) that

$$\begin{aligned} P_1(d_0) = P_2(d_0) = \ldots = P_{n-1}(d_0) = 0. \end{aligned}$$

(2.2)

Since, by (1.3), $P_1 (d_0) = c_0 d_0 + c_1$, we get

$$\begin{aligned} d_0 = -\frac{c_1}{c_0}. \end{aligned}$$

(2.3)

Further, since, by (1.1),

$$\begin{aligned} P_k (x) = \frac{k!}{(n-1)!} P_{n-1}^{(n-1-k)} (x), \quad k=1,\ldots ,n-1, \end{aligned}$$

(2.4)

we infer that $d_0$ is a root of $P_{n-1}(x)$ of multiplicity $(n-1)$. Thus, in view of (2.3), we have $P_{n-1}(x) = c_0 \left( x + c_1/c_0\right) ^{n-1}$, which implies

$$\begin{aligned} P_n (x) = c_0 \left( x + \frac{c_1}{c_0}\right) ^{n} + C \quad \text {with }\,\, C = P_n \left( -\frac{c_1}{c_0}\right) . \end{aligned}$$

(2.5)

First, if  $n\ge 7$ is even , then, by (2.5), one can easily find the nontrivial decomposition $P_{n} (x) = Q(R(x))$ with

$$\begin{aligned} Q(x)= c_0 x^{2} + C, \quad \text {and }\,\,R(x)=\left( x + \frac{c_1}{c_0}\right) ^{n/2}. \end{aligned}$$

(2.6)

Since $n \ge 7$, this nontrivial decomposition is obviously not equivalent to the one in (1.4), contradicting that $\left\{ P_n (x)\right\} _{n \ge 0}$ is of special type.

Now, let $n \ge 7$ be an odd positive integer. If $n$ is composite, then any divisor $v$ of $n$ with $1< v < n$ leads to a nontrivial decomposition

$$\begin{aligned} P_{n} (x) = c_0 \left( \left( x + \frac{c_1}{c_0}\right) ^{v}\right) ^{n/v} + C, \end{aligned}$$

(2.7)

which again contradicts that $\left\{ P_n (x)\right\} _{n \ge 0}$ is of special type (and in this case $P_{n}(x)$ is indecomposable). If $n$ is a prime, then derivating both sides of (2.5) we obtain

$$\begin{aligned} P_{n -1} (x) = c_0 \left( x - \frac{1}{2}\right) ^{n -1}, \end{aligned}$$

(2.8)

where of course the exponent $n-1$ is even. Similarly as above, this leads to a nontrivial decomposition not equivalent to (1.4) and thus to a contradiction. $\square$

Proposition 2.4

Suppose that $\left\{ P_n (x)\right\} _{n \ge 0}$ is an Appell sequence which satisfies (1.6). Then the polynomial $P_n (d_1 x + d_0)$ is not of the form $e_1 D_{\mu } (x,\delta ) + e_0$, where $D_{\mu } (x,\delta )$ the $\mu$-th Dickson polynomial with $\mu >4$, $\delta \in \mathbb {Q}^*$.

Proof

Suppose that the Appell sequence $\left\{ P_n (x)\right\} _{n \ge 0}$ satisfies (1.6), and that we have

$$\begin{aligned} P_n (d_1 x + d_0) = e_1 D_{\mu } (x,\delta ) + e_0. \end{aligned}$$

(2.9)

Clearly, $n=\mu$. Comparing the leading coefficients of both sides we get

$$\begin{aligned} d_1^n c_0 = e_1, \end{aligned}$$

(2.10)

where the numbers $c_k \ (k \ge 0)$ are defined in (1.3). Similarly, from (1.2) and the equality of the coefficients of $x^{n-1}$ on both sides we obtain

$$\begin{aligned} n d_1^{n-1} P_1(d_0) = 0, \end{aligned}$$

(2.11)

which implies

$$\begin{aligned} d_0 = -\frac{c_1}{c_0}. \end{aligned}$$

(2.12)

Again, by (1.2), comparing the coefficients of $x^{n-2}$ gives

$$\begin{aligned} \left( {\begin{array}{c}n\\ 2\end{array}}\right) d_1^{n-2} P_2(d_0) = -e_1 n \delta , \end{aligned}$$

(2.13)

whence, together with (2.10) it follows that

$$\begin{aligned} d_1^2 = -\frac{(n-1)P_2(d_0)}{2c_0 \delta } \end{aligned}$$

(2.14)

Now we compare the coefficients of $x^{n-4}$ on both sides of (2.9) and we obtain

$$\begin{aligned} \left( {\begin{array}{c}n\\ 4\end{array}}\right) d_1^{n-4} P_4(d_0) = \frac{e_1 n (n-3) \delta ^2}{2}, \end{aligned}$$

(2.15)

which along with (2.10) leads to

$$\begin{aligned} d_1^4 = \frac{(n-1)(n-2) P_4(d_0)}{12 c_0 \delta ^2}. \end{aligned}$$

(2.16)

After substituting (2.14) into (2.16), we obtain

$$\begin{aligned} 3 (n-1) P_2(d_0)^2 = (n-2) c_0 P_4(d_0), \end{aligned}$$

(2.17)

whence, together with (2.12) it follows that

$$\begin{aligned} n = \frac{3 P_2 (-c_1 / c_0)^2 - 2 c_0 P_4 (-c_1 / c_0)}{3 P_2 (-c_1 / c_0)^2 - c_0 P_4 (-c_1 / c_0)}. \end{aligned}$$

(2.18)

This is a contradiction by (1.6). $\square$

We note that Proposition 2.4 is a common generalization of Lemma 5.3 in [6], Lemma 2.4 in [3], and of the second statement of Lemma 12 in [11].

3 Proof of Theorem 1.1

Let $g(x) \in \mathbb {Q}[x]$ with $\deg g(x) \ge 3$. Suppose that equation (1.5) has infinitely many integer solutions $x,y$ with an Appell sequence $\left\{ P_n (x)\right\} _{n \ge 0}$ of special type satisfying (1.6) and with $n\ge 7$. Then by Proposition 2.1 it follows that there exist $\lambda (x), \mu (x), \varphi (x) \in \mathbb {Q}[x]$, $\deg \lambda (x) = \deg \mu (x) =1$ such that

$$\begin{aligned} P_n(x)=\varphi (f_1(\lambda (x))) \quad \text { and } \,\, g(x)=\varphi (g_1(\mu (x))), \end{aligned}$$

(3.1)

where $(f_1(x), g_1(x))$ is a standard pair over $\mathbb {Q}$.

Let $\lambda ^{-1}(x) = a_1x+a_0, \mu ^{-1}(x) = b_1x+b_0$, where $a_0, a_1, b_0, b_1 \in \mathbb {Q}$ with $a_1b_1\ne 0$. Then we can rewrite (3.1) as

$$\begin{aligned} P_n(a_1x+a_0)=\varphi (f_1(x)) \quad \text { and }\,\, g(b_1x+b_0)=\varphi (g_1(x)), \end{aligned}$$

(3.2)

Since $P_n(x)$ is of special type and $\deg P_n (x) = n$, we obtain that

$$\begin{aligned} \deg \varphi (x) \in \left\{ 1, \frac{n}{2}, n\right\} . \end{aligned}$$

3.1 The case $\deg \varphi (x) = n$

If we assume that $\deg \varphi (x) = n$, then by (3.1), we have $\deg f_1(x) = 1$. Thus $P_n (x) = \varphi (t(x))$, where $t(x) \in \mathbb {Q}[x]$ is a linear polynomial. Clearly, $t^{-1}(x) \in \mathbb {Q}[x]$ is also linear. By (3.1), we obtain $P_n(t^{-1}(x)) = \varphi (t(t^{-1}(x))) = \varphi (x)$. Hence

$$\begin{aligned} g(x) = \varphi (g_1(\mu (x))) = P_n(t^{-1}(g_1(\mu (x)))) = P_n (q(x)), \end{aligned}$$

(3.3)

where $q(x) = t^{-1}(g_1(\mu (x)))$. So, if, in our case, equation (1.5) has infinitely many solutions, then $g(x)$ is of the form as in Theorem 1.1 (i).

3.2 The case $\deg \varphi (x)=1$

Let $\varphi (x) = \varphi _1 x + \varphi _0$, where $\varphi _1,\varphi _0 \in \mathbb {Q}$ and $\varphi _1 \ne 0$. We now study the five kinds of standard pairs.

In view of our assumptions on $n$ and $\deg g(x)$, it follows that the standard pair $(f_1(x),g_1(x))$ cannot be of the second or fifth kind.

If it is of the third or fourth kind, we then have $P_n (a_1x+a_0) = e_1D_{\mu }(x,\delta ) + e_0$ with $e_0 \in \mathbb {Q}, e_1,\delta \in \mathbb {Q}^*$. This contradicts Proposition 2.4.

If $(f_1(x),g_1(x))$ is a standard pair of the first kind, then we have either

• (I) $P_n (a_1x+a_0) = \varphi _1 x^q + \varphi _0$, or

• (II) $P_n (a_1x+a_0) = \varphi _1 \alpha x^p \nu (x)^q + \varphi _0$, where $0 \le p < q, (p,q)=1$ and $p + \deg \nu (x) > 0$.

The first case (I) is impossible by Proposition 2.3 since $n \ge 7$ by assumption.

Let us now consider the second case (II). Then we have $g(x) = \varphi _1 \mu (x)^q + \varphi _0 = \varphi (\mu (x)^q)$, where $q \ge 3$ and $\mu (x) \in \mathbb {Q}[x]$ is linear, which is case (ii) of Theorem 1.1.

3.3 The case $\deg \varphi (x)=n/2$

Clearly, $n$ is then even, and from (3.1) we observe that $\deg f_1 (x) = 2$. Hence it follows that, in (3.1), $(f_1(x),g_1(x))$ cannot be a standart pair of the fifth kind. Further, we obtain a nontrivial decomposition of $P_n (x)$, which, since $P_n(x)$ is of special type, implies that there exists a linear polynomial $\ell (x)= \ell _1 x + \ell _0$ over $\mathbb {Q}$ such that

$$\begin{aligned} \varphi (x) = \widehat{P}_{n/2} ( \ell (x) )\quad \text { and } \,\, \ell (f_1(\lambda (x))) = \left( x - \frac{1}{2}\right) ^2. \end{aligned}$$

(3.4)

Again, we study the remaining kinds of standard pairs.

First, we consider the case when, in (3.1), $(f_1(x),g_1(x))$is a standard pair of the first kind. If $f_1 (x) = x^t$, then by $\deg f_1 (x) = 2$, we have $(f_1(x),g_1(x)) = (x^2, \alpha x p(x)^2)$. Putting $\lambda (x) = \lambda _1 x + \lambda _0$, (3.4) takes the form $\ell ((\lambda _1 x + \lambda _0)^2) = \left( x - 1/2\right) ^2$, whence an easy calculation gives $\ell (x) = x / \lambda _1^2$. Substituting this into (3.1), we obtain

$$\begin{aligned} g(x) = \widehat{P}_{n/2} ( \ell (g_1(\mu (x))) ) = \widehat{P}_{n/2} \left( \frac{\alpha \mu (x) p(\mu (x))^2}{\lambda _1^2} \right) \end{aligned}$$

(3.5)

So $g(x)$ is of the form (iv) with $\delta (x) = \alpha \mu (x) / \lambda _1^2$ and $q(x) = p(\mu (x))$.

In the switched case $(f_1 (x), g_1(x)) = (\alpha x^r p(x)^t,x^t)$, where $0 \le r<t, (r,t)=1$ and $r+ \deg p(x) >0, \deg f_1 (x) = 2$ implies that one of the following cases occurs:

• (A) $r=0, \ t=1$ and $\deg p(x) = 2$, or

• (B) $r=2, \ t>2$ is odd and $p(x)$ is a constant polynomial.

In case (A) we have $g_1 (x) = x$, whence from (3.1) and (3.4) we obtain

$$\begin{aligned} g(x) = \widehat{P}_{n/2} ( \ell (g_1(\mu (x))) ) = \widehat{P}_{n/2} ( \ell (\mu (x)) ) = \widehat{P}_{n/2} ( \delta (x) q(x)^2), \end{aligned}$$

(3.6)

where $\delta (x) = \ell (\mu (x))$ and $q(x) \equiv 1$. Thus $g(x)$ is again of the form (iv).

In the second case (B), we can write $f_1(x) = \beta x^2$, with $\beta = \alpha p(x)^t \in \mathbb {Q} \setminus \left\{ 0\right\}$. Substituting this into (3.4), we deduce that $\ell (x) = x / (\beta \lambda _1^2)$, whence, by (3.1), we get

$$\begin{aligned} g(x) = \widehat{P}_{n/2} ( \ell (g_1(\mu (x))) ) = \widehat{P}_{n/2} \left( \frac{\mu (x)^t}{\beta \lambda _1^2} \right) = \widehat{P}_{n/2} ( c \delta (x)^t ), \end{aligned}$$

(3.7)

where $c= 1/(\beta \lambda _1^2), \ \delta (x) = \mu (x)$ and $t>2$ is odd. This is option (v) in Theorem 1.1.

Next let $(f_1(x), g_1(x))$, in (3.1), be a standard pair of the second kind. If $(f_1(x),g_1(x)) = (x^2,(\alpha x^2 + \beta ) v(x)^2)$, then a calculation from (3.4) yields $\ell (x) = x/\lambda _1^2$, and by (3.1) we have

$$\begin{aligned} g(x)&= \widehat{P}_{n/2} ( \ell (g_1(\mu (x))) ) = \nonumber \\&= \widehat{P}_{n/2} \left( \frac{(\alpha \mu (x)^2 + \beta ) v(\mu (x))^2}{\lambda _1^2} \right) = \widehat{P}_{n/2} ( (\alpha \delta (x)^2 + \beta ) q(x)^2 ), \end{aligned}$$

(3.8)

where $\delta (x) = \mu (x)$ and $q(x) = v(\mu (x))/ \lambda _1$. So we are led to option (vi) of our theorem.

In the switched case $(f_1(x),g_1(x)) = ((\alpha x^2 + \beta ) v(x)^2, x^2)$, since $\deg f_1 (x) = 2$, $v(x)$ is a constant polynomial and

$$\begin{aligned} g(x) = \widehat{P}_{n/2} ( \ell (g_1(\mu (x))) ) = \widehat{P}_{n/2} ( ( \ell _1 \mu (x)^2 + \ell _0 ) q(x)^2 ), \end{aligned}$$

(3.9)

where $q(x) \equiv 1$. Thus, we arrived again at option (vi) with $\delta (x) = \mu (x)$ and $a = \ell _1, b= \ell _0$.

Now, if the standard pair $(f_1(x),g_1(x))$ is of the third kind over $\mathbb {Q}$, then $(f_1(x),g_1(x)) = (D_2 (x, \alpha ^t), D_t (x, \alpha ^2))$ with $t$ being odd. Let us substitute $f_1 (x) = x^2 - 2 \alpha ^t$ into (3.4) to deduce that $\ell (x) = (x + 2 \alpha ^t)/ \lambda _1^2$, whence

$$\begin{aligned} g(x) = \widehat{P}_{n/2} ( \ell (g_1(\mu (x))) ) = \widehat{P}_{n/2} \left( \frac{D_t (\mu (x), \alpha ^2) + 2\alpha ^t}{\lambda _1^2} \right) . \end{aligned}$$

(3.10)

It follows from Proposition 2.2 that $-2 \alpha ^t/ \lambda _1^2$ is an extremum of the polynomial $D_t (\mu (x)$, $\alpha ^2)/ \lambda _1^2$, which is of $P$-type $(1, 2, \ldots , 2)$ as $t$ is odd. Hence $(D_t (\mu (x), \alpha ^2) + 2 \alpha ^t)/ \lambda _1^2 = \delta (x) q(x)^2$ for some $\delta (x), q(x) \in \mathbb {Q}[x]$ with $\deg \delta (x) = 1$. We deduce, that $g(x)$ is of the form (iv).

Finally, consider the case when $(f_1(x),g_1(x))$ is a standard pair of the fourth kind over $\mathbb {Q}$. Then

$$\begin{aligned} (f_1(x),g_1(x)) = \left( \frac{D_2 (x, \alpha )}{\alpha }, \frac{D_t (x, \beta )}{\beta ^{(t/2)}}\right) , \end{aligned}$$

with an even $t$. Substituting this into (3.4), an easy calculation yields $\ell (x) = (\alpha x + 2 \alpha )/ \lambda _1^2$, whence, by (3.1), we obtain

$$\begin{aligned} g(x) = \widehat{P}_{n/2} ( \ell (g_1(\mu (x))) ) = \widehat{P}_{n/2} \left( \frac{\alpha \beta ^{-t/2} D_t (\mu (x), \beta ) + 2\alpha }{\lambda _1^2} \right) . \end{aligned}$$

(3.11)

Now from Proposition 2.2 we infer that

$$\begin{aligned} -\frac{2 \beta ^{t/2} \alpha \beta ^{-t/2} }{\lambda _1^2} = -\frac{2\alpha }{\lambda _1^2} \end{aligned}$$

is one of the two extrema of the polynomial $\alpha \beta ^{-t/2} D_t (\mu (x), \beta )/ (\lambda _1^2)$ and it is of $P$-type $(2, 2, \ldots , 2)$, as $t$ is even. Therefore we have

$$\begin{aligned} \frac{\alpha \beta ^{-t/2} D_t (\mu (x), \beta ) + 2 \alpha }{\lambda _1^2} = q(x)^2 \end{aligned}$$

for some $q(x) \in \mathbb {Q}[x]$. Thus $g(x)$ is of the form (iii), which completes the proof.