Finsler metrics are direct generalizations of Riemannian metrics such that the quadratic Riemannian indicatrices in the tangent spaces of a manifold are replaced by more general convex bodies as unit spheres. A linear connection on the base manifold is called compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called generalized Berwald manifolds [11]. Compatible linear connections are the solutions of the so-called compatibility equations containing the components of the torsion tensor as unknown quantities. Although there are some theoretical results for the solvability of the compatibility equations  (monochromatic Finsler metrics [1], extremal compatible linear connections and algorithmic solutions [9]), it is very hard to solve in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique.  Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor. It is proved [6] (see also [7]) that such a compatible linear connection must be uniquely determined.
The original proof is based on averaging in the sense that the 1-form in the decomposition of the torsion tensor can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. In what follows we present a new proof for the unicity result by using linear algebra and some basic facts about convex bodies. We present an explicit formula for the solution without integration. The method has a new contribution to the problem as well:  necessary conditions of the solvability are formulated in terms of intrinsic equations without unknown quantities. They are sufficient if and only if the solution depends only on the position.

§ COMPATIBILITY EQUATIONS IN FINSLER GEOMETRY

Let $M$ be a smooth connected manifold with a local coordinate system $u^1, \ldots, u^n$. The induced local coordinate system on the tangent manifold $TM$ consists of the functions $x^1, \ldots, x^n$ and $y^1, \ldots, y^n$ given by $x^i(v):=u^i\circ \pi (v)=u^i(p)=:p^i$, where  $\pi \colon TM \to M$ is the canonical projection and $y^i(v)=v(u^i)$, $i=1, \ldots, n$. Throughout the paper, we will use the shorthand notations

\begin{equation}
\partial_i := \dfrac{\partial}{\partial x^i} \hspace{1cm} \text{and} \hspace{1cm} \dot{\partial}_i:= \dfrac{\partial}{\partial y^i}.
\end{equation}

A Finsler metric [4] on a manifold is a smoothly varying family of  Minkowski norms in the tangent spaces. It is a direct generalization of Riemannian metrics, with inner products (quadratic indicatrices) in the tangent spaces replaced by Minkowski norms (smooth strictly convex bodies).

A Finsler metric is a non-negative continuous function $F\colon TM\to \mathbb{R}$ satisfying the following conditions:  $\displaystyle{F}$ is smooth on the complement of the zero section (regularity), $\displaystyle{F(tv)=tF(v)}$ for all $\displaystyle{t> 0}$ (positive homogeneity), $F(v)= 0$ if and only if $v={\bf 0}$  (definiteness) and the Hessian $\displaystyle{g_{ij}=\dot{\partial}_{i}\dot{\partial}_{j}E}$ of the energy function $E=F^2/2$ is positive definite at all non-zero elements $\displaystyle{v\in T_pM}$ (strong convexity). The pair $(M,F)$ is called a Finsler manifold.

On a Riemannian manifold we obviously have compatible linear connections in the sense that the induced parallel transports preserve the Riemannian length of tangent vectors (metric linear connections). Following the classical Christoffel process it is clear that such a linear connection is uniquely determined by the torsion tensor. In contrast to the Riemannian case, non-Riemannian Finsler manifolds admitting compatible linear connections form a special class of spaces in Finsler geometry. They are called generalized Berwald manifolds [11]. It is known that some Finsler manifolds do not admit any compatible linear connections because of topological constraints, some have infinitely many compatible linear connections and it can also happen that the compatible linear connection is uniquely determined [8], see also  [10].

A linear connection is compatible with the Finsler metric if the induced parallel transports preserve the Finslerian length of tangent vectors. Finsler manifolds admitting compatible linear connections are called  generalized Berwald manifolds.

In terms of local coordinates, equations

\begin{equation}
\label{ceq_christoffel}
X_i^h F := \partial_i F-y^j \left( {\Gamma}^k_{ij}\circ \pi \right) \dot{\partial}_k F=0 \hspace{1cm} (i=1,\dots,n)
\end{equation}

form necessary and sufficient conditions for a linear connection $\nabla$ to be compatible with the Finsler metric $F$. Equations (<ref>) are called compatibility equations or CEQ for short. The fundamental result of generalized Berwald manifold theory  states that a compatible linear connection $\nabla$ is always Riemann metrizable [5], i.e. $\nabla$ must be a metric linear connection with respect to a Riemannian metric $\gamma$. Such a Riemannian metric can be given by integration of $g_{ij}$ on the indicatrix hypersurfaces [5], see also [2] and [3]. It is the so-called averaged Riemannian metric. Therefore CEQ can be reformulated by replacing the Christoffel symbols  $\Gamma_{ij}^k$ by the torsion tensor components [9], see also  [10]. Using the horizontal vector fields

\[ X_i^{h*}:=\partial_i -y^j \left( {\Gamma}^{k*}_{ij}\circ \pi \right)  \dot{\partial}_k \hspace{1cm} (i=1,\dots,n), \]

where ${\Gamma}^{k*}_{ij}$ are the Christoffel symbols of the Lévi-Civita connection of the averaged Riemannian metric $\gamma$, CEQ takes the form

\begin{equation}
y^j \left(T^{l}_{jk}\gamma^{kr}\gamma_{il}+T^{l}_{ik}\gamma^{kr}\gamma_{jl}-T_{ij}^r\right) \dot{\partial}_r F=-2X_i^{h^*}F\hspace{1cm} (i=1,\dots,n).
\end{equation}

The unknown quantities  $T_{ab}^c$ are the torsion tensor components  of the compatible linear connection.

In what follows we are going to use normal coordinates with respect to the Riemannian metric $\gamma$ around a given point $p\in M$. The coordinate vector fields $\partial/\partial u^1, \dots, \partial/\partial u^n$ form an orthonormal basis in $T_p M$, i.e. $\gamma_{ij}(p)=\delta_{ij}$ and ${\Gamma}^{k*}_{ij}(p)=0$. Therefore $X_i^{h*}(v)=\partial_i(v)$ for any $v\in T_pM$ and CEQ takes the form

\begin{equation}
\label{CEQ-tors2}
\sum_{a<b,c} \sigma_{ab;i}^c T_{ab}^{c} = -2\partial_i F \hspace{1cm} (i=1, \dots, n),
\end{equation}

where the coefficients are

\begin{equation}
\label{CEQ-coeff2}
\sigma_{ab;i}^c := \delta_i^a f_{cb} + \delta_i^b f_{ac} + \delta_i^c f_{ab}, \hspace{1cm} f_{ij} := y^i \dot{\partial}_j F - y^j \dot{\partial}_i F.
\end{equation}

If none of the indices $a,b,c$ are equal to $i$ then $\sigma_{ab;i}^c=0$. Otherwise the table shows the possible cases, where indices are separated according to their values (equal indices are put into the same cell and different cells contain different values).

=1pt  |c||c|c|c||l|
3c||indices     the coefficients

1.     $i=a$     $b$     $c$     $\sigma_{ib;i}^{c}=f_{cb}$

2.     $i=a$     $b=c$           $\sigma_{ib;i}^{b}=0$

3.     $i=b$     $a$     $c$     $\sigma_{ai;i}^{c}=f_{ac}$

4.     $i=b$     $a=c$           $\sigma_{ai;i}^{a}=0$

5.     $i=c$     $a$     $b$     $\sigma_{ab;i}^{i}=f_{ab}$

6.     $i=a=c$     $b$           $\sigma_{ib;i}^{i}=2f_{ib}$

7.     $i=b=c$     $a$           $\sigma_{ai;i}^{i}=2f_{ai}$

Therefore the $i$-th compatibility equation at the point $p$ is

\begin{equation} \label{CEQ-general}
\sideset{}{'}\sum_{a} 2f_{ia} T_{ia}^i + \sideset{}{'}\sum_{a<b} f_{ab} \left( T_{ib}^a + T_{ab}^i + T_{ai}^b \right)  = -2\, \partial_i F,
\end{equation}

where the primed summation means summing for $a \neq i$ in the first one, and in the second one, $a$ and $b$ where $i\neq a, b$.

§ THE GEOMETRY OF THE TANGENT SPACES

The following table shows a panoramic view about the geometric structures of the tangent space $T_pM$ due to the simultaneously existing Finsler and Riemannian metrics.

Finsler structure           Riemannian structure

Minkowski norm     metric on $T_p M$     Euclidean norm and inner product

Finslerian spheres $F_p(\lambda)$     level sets of $\lambda\in \rr_+$     Euclidean spheres $R_p(\lambda)$

$\mathcal{F}_v$     tangent hyperplanes of level sets at $v\in T_pM$     $\mathcal{R}_v$

$\mathcal{LF}_v:= \mathcal{F}_v - v$      linear tangent hyperplanes at $v\in T_pM$     $\mathcal{LR}_v := \mathcal{R}_v - v$

$G := \mathrm{grad} \, F = [ \dot{\partial}_1 F, \dots,  \dot{\partial}_n F]$     normal vector fields (w.r.t. $\gamma$)     $C := [y^1, \dots, y^n]$

Both gradient vector fields $G$ and $C$ are nonzero everywhere on $T_p^{\circ} M := T_p M \backslash \{ \textbf{0} \}$. For every element $v \in T_p^{\circ} M$ the tangent hyperplanes of the Finslerian and the Riemannian (Euclidean) spheres passing through $v$ can be related as follows.

* If $\mathcal{F}_v=\mathcal{R}_v$, i.e. $G_v \parallel C_v$, we call the point $v$ a vertical contact point of the metrics.

* If $\mathcal{F}_v \neq \mathcal{R}_v$, i.e. $G_v$ and $C_v$ are linearly independent, then the intersection $\mathcal{LF}_v \cap \mathcal{LR}_v$ is the orthogonal complement of $\mathrm{span}(C_v, G_v)$, and thus $\mathcal{F}_v \cap \mathcal{R}_v$ is an affine subspace of dimension $n-2$.

Let us define the vector field

\begin{equation} \label{H-def}
H_v := \left[ \partial_1 F(v), \dots, \partial_n F(v) \right].
\end{equation}

An element $v \in T_p^{\circ} M$ is a horizontal contact point of the metrics if $H_v$ is the zero vector. It can be easily seen  that if $T_p M$ is a vertical contact tangent space, i. e. all of its non-zero elements are vertical contact, then the Finsler metric is a scalar multiple of $\gamma$ at the point $p\in M$. The quadratic indicatrix of a generalized Berwald metric at a single point means quadratic indicatrices at all points of the (connected) base manifold because the tangent spaces are related by linear isometries due to the parallel transports with respect to the compatible linear connection with the Finsler metric. Therefore such a generalized Berwald manifold reduces to a Riemannian manifold.  At a horizontal contact point $v\in T_pM$, equations of CEQ are homogeneous. If $T_p M$ is a horizontal contact tangent space, i. e. all of its non-zero elements are horizontal contact, then $T \equiv 0$ is a solution of CEQ at $p\in M$.

§.§ A useful family of vector fields
Let us define the vector fields

\begin{equation} \label{rowvector}
f_i(v) := \left[ f_{i1}(v), f_{i2}(v), \dots, f_{in}(v) \right]^T \hspace{1cm} (i=1,\dots, n)
\end{equation}

on $T_p^{\circ} M$ to help in proving some elementary properties and solving CEQ.

For any $v \in T_p^{\circ} M$, we have $\mathrm{span}(f_1(v), \dots, f_n(v)) \subseteq  \mathrm{span}(G_v, C_v)$.

Observe that $f_i$ can be written as

\begin{equation} \label{rowvector-lincomb}
f_i = \begin{bmatrix} f_{i1} \\ f_{i2} \\ \vdots \\ f_{in}\end{bmatrix} =
\begin{bmatrix} y^i \dot{\partial}_1 F - y^1  \dot{\partial}_i F \\ y^i  \dot{\partial}_2 F - y^2  \dot{\partial}_i F \\ \vdots \\ y^i  \dot{\partial}_n F - y^n  \dot{\partial}_i F \end{bmatrix} =
y^i \begin{bmatrix}  \dot{\partial}_1 F \\  \dot{\partial}_2 F \\ \vdots \\  \dot{\partial}_n F \end{bmatrix} -  \dot{\partial}_i F \begin{bmatrix}
y^1 \\ y^2 \\ \vdots \\ y^n \end{bmatrix},
\end{equation}

i.e. $f_i = y^i \cdot G -  \dot{\partial}_i F \cdot C$.

At a vertical contact point $v \in T_p^{\circ} M$,  $f_i(v)=0$ and  CEQ takes the form $0=\partial_i F(v)$ $(i=1, \ldots, n)$.

If $v$ is vertically contact, then $G_v = \lambda C_v$ for some nonzero $\lambda \in \rr$. Substituting into (<ref>), we get

\[ f_i(v) = v^i \cdot \lambda C_v - \lambda v^i \cdot C_v = 0 \hspace{1cm} (i=1, \dots, n), \]

i.e. the coordinates of $f_i$ are all zero at $v$ and the reformulation (<ref>) of CEQ implies the statement.

In order for CEQ to have a solution, all vertically contact points must be horizontal contact.

At a not vertically contact $v \in T_p^{\circ} M$ there are indices such that $f_{ij}(v)\neq 0$ and the vectors $f_i$ and $f_j$ are linearly independent over some neighborhood $U$ of $v$ in $T_pM$.

Suppose that all the $f_{ij}(v)$, and consequently, all the vectors $f_i(v)$ are zero. Since $v$ is not vertically contact, $G_v$ and $C_v$ are linearly independent. In particular, neither of them is the zero vector, so one of their coordinates is nonzero, meaning that for some index $i$, (<ref>) gives the zero vector as a linear combination of the independent vectors $C_v$ and $G_v$ with nonzero coefficients. This is a contradiction, so there must be an $f_{ij}(v)$, and thus two vectors $f_i(v)$ and $f_j(v)$ different from zero. Since the matrix

\[ \begin{bmatrix} f_i \\ f_j \end{bmatrix} =
\begin{bmatrix} f_{i1} & \dots & 0 & \dots & f_{ij} & \dots & f_{in} \\ f_{j1} & \dots & f_{ji} & \dots & 0 & \dots & f_{jn}\\ \end{bmatrix} \]

has rank 2 at $v$ (choose the $i$-th and $j$-th columns), they are linearly independent at $v$ and the same is true at the points of some adequately small neighborhood of $v$ in  $T_pM$ by a continuity argument.

At a point $v \in T_p^{\circ} M$ the vectors defined by (<ref>) span the subspace

\[ \mathrm{span}(f_1(v), \dots, f_n(v)) = \left\{\begin{array}{cl}
\{ \bm{0} \} & \text{if} \ v \ \text{is vertically contact}, \\
\mathrm{span}(G_v, C_v) &  \text{if} \ v \ \text{is not vertically contact}.
\end{array}\right. \]

For any $v \in T_p^{\circ} M$, we have $\mathrm{span}(f_1(v), \dots, f_n(v)) \subseteq \mathrm{span}(G_v, C_v)$ by Lemma <ref>. If $v$ is vertically contact, all the vectors $f_i(v)$ are zero according to Lemma <ref>. If not, there are 2 independent vectors among them according to Lemma <ref>, thus generating the whole $\mathrm{span}(G_v, C_v)$.

At a not vertically contact $v \in T_p^{\circ} M$, let us choose indices $i \neq j$ such that $f_{ij}(v)\neq 0$. Then $(f_i, f_j)$ is a basis of $\mathrm{span}(G, C)$  over some neighborhood $U$ of $v$ in $T_pM$ and

\begin{equation}\label{rowvector-lincomb2}
f_k = \dfrac{f_{kj}}{f_{ij}} \cdot f_i + \dfrac{f_{ik}}{f_{ij}} \cdot  f_j \hspace{1cm} (k=1, \dots, n)
\end{equation}

at any point of $U$.

By Lemma <ref>, we know that $(f_i, f_j)$ is a basis of $\mathrm{span}(G, C)$ at the points of $U$. Let us choose an index $k \in \{ 1, \dots, n\}$ and write $f_k = \lambda_1 f_i + \lambda_2 f_j$. By (<ref>), we can write that

\[f_k= y^k \cdot G - \dot{\partial}_k F \cdot C =  \lambda_1 \left( y^i \cdot G - \dot{\partial}_i F \cdot C \right) + \lambda_2 \left( y^j \cdot G - \dot{\partial}_j F \cdot C \right). \]

By comparing the coefficients in the basis $(G, C)$,

\[ \begin{bmatrix} y^k \\ \dot{\partial}_k F \end{bmatrix} = \begin{bmatrix} y^i & y^j \\ \dot{\partial}_i F & \dot{\partial}_j F \end{bmatrix} \cdot \begin{bmatrix} \lambda_1 \\ \lambda_2
\end{bmatrix} \]

and, consequently,

\[ \begin{bmatrix} \lambda_1 \\ \lambda_2
\end{bmatrix} = \dfrac{1}{f_{ij}} \begin{bmatrix} \dot{\partial}_j F & -y^j \\ -\dot{\partial}_i F & y^i \end{bmatrix} \cdot \begin{bmatrix} y^k \\ \dot{\partial}_k F \end{bmatrix} = \dfrac{1}{f_{ij}} \begin{bmatrix} y^k \dot{\partial}_j F - y^j \dot{\partial}_k F \\ y^i \dot{\partial}_k F - y^k \dot{\partial}_i F \end{bmatrix}. \qedhere \]

§ THE SEMI-SYMMETRIC COMPATIBLE LINEAR CONNECTION AND ITS UNICITY

Although there are some theoretical results for the solvability of the compatibility equations  (monochromatic Finsler metrics [1], extremal compatible linear connections and algorithmic solutions [9]), it is very hard to solve in general because compatible linear connections may or may not exist on a Finsler manifold and may or may not be unique.  Therefore special cases are of special interest. One of them is the case of the so-called semi-symmetric compatible linear connection with decomposable torsion tensor.

A linear connection is called semi-symmetric if its torsion tensor can be written as

\begin{equation}\label{semi-symm-tors}
T(X,Y) = \rho(Y) X - \rho(X)Y
\end{equation}

for some differential 1-form $\rho$ on the base manifold.

It is proved [6] that a semi-symmetric compatible linear connection must be uniquely determined.

[6] A non-Riemannian Finsler manifold admits at most one semi-symmetric compatible linear connection.

The original  proof is based on averaging in the sense that the 1-form $\rho$  can be expressed by integrating differential forms on the tangent manifold over the Finslerian indicatrices. The integral formulas are very difficult to compute in practice. In what follows we present a new proof for the unicity result by using linear algebra and some basic facts about convex bodies. We present an explicit formula for the solution without integration. The method has a new contribution to the problem as well:  necessary conditions of the solvability are formulated in terms of intrinsic equations without unknown quantities. They are sufficient if and only if the solution depends only on the position.

§.§ The proof of Theorem <ref>

$$T(\partial/\partial u^i, \partial/\partial u^j) =  \rho(\partial/\partial u^j) \partial/\partial u^i - \rho(\partial/\partial u^i) \partial/\partial u^j=$$

$$= \rho_j \partial/\partial u^i - \rho_i \partial/\partial u^j = \left( \delta_i^k \rho_j - \delta_j^k \rho_i \right) \partial/\partial u^k,$$

the torsion components are

\begin{equation} \label{semi-symm-torscomp}
T_{ij}^k = \delta_i^k \rho_j - \delta_j^k \rho_i.
\end{equation}

In particular, all torsion components with 3 different indices are zero, and

\[ T_{ij}^i = \delta_i^i \rho_j - \delta_j^i \rho_i =\rho_j \hspace{1cm }(j \neq i). \]

Substituting the torsion components into the general form (<ref>) of CEQ at a point $p$, it takes the (matrix) form

\begin{equation} \label{CEQ}
\begin{tabu} to .5\textwidth {|ccccc|c|}
\hline
\rho_1 & \rho_2 & \rho_3 & \cdots & \rho_n & \text{RHS} \\
\hline
0 & f_{12} & f_{13} & \cdots & f_{1n} & -\partial_1 F \\
f_{21} & 0 & f_{23} & \cdots & f_{2n} & -\partial_2 F \\
f_{31} & f_{32} & 0 & \cdots & f_{3n} & -\partial_3 F \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
f_{n1} & f_{n2} & f_{n3} & \cdots & 0 & -\partial_n F \\
\hline
\end{tabu}}
\end{equation}

The problem is to solve (<ref>) for $\rho_1, \dots, \rho_n$, considered as the coordinates of a vector $\rho \in T_p M$, as $v$ ranges over $T_p^{\circ} M$. To prove Theorem <ref> it is enough to consider the homogeneous version  H-CEQ with vanishing right hand side of  (<ref>). We are going to verify that the only solution of H-CEQ is $\rho_1=\dots=\rho_n=0$. Since the rows of the matrix on the left hand side are exactly the vectors $f_1, \dots, f_n$ defined in (<ref>), solving H-CEQ at a fixed element $v$ means finding the orthogonal complement of $\mathrm{span}(f_1(v), \dots, f_n(v))$. By Corollary <ref>,

* it is $T_p M$ for any vertically contact element $v$,

* it is the orthogonal complement of $\mathrm{span}(G_v, C_v)$ for any not vertically contact element $v$, i.e. the intersection $\mathcal{LF}_v \cap \mathcal{LR}_v$ of the linear tangent hyperplanes of the Finslerian and Riemannian spheres.

The solution of H-CEQ at the point $p$ is the intersection of all the solution spaces as the element $v$ ranges over $T_p^{\circ} M$. Note that the homogeneity of the coefficients imply that it is enough to consider the intersection of all the solution spaces as the element $v$ ranges over the Finslerian (or the Riemannian) unit sphere.

* If all the elements of $T_p M$ are vertically contact and the Finsler manifold admits a compatible (semi-symmetric) linear connection $\nabla$, then it is a Riemannian manifold because the linear isometries via the parallel transports with respect to $\nabla$ extend the quadratic Finslerian (esp. Riemannian) indicatrix at the point $p$ to the entire (connected) manifold.

* If there is a not vertically contact element $v$, then, by a continuity argument, we can consider a neighborhood $U\subseteq T_pM$ containing not vertically contact elements. For the solution vector $\rho$ we have

\[ \rho \in \bigcap_{v \in U} \left( \mathcal{LF}_v \cap \mathcal{LR}_v \right) \subseteq \big(\bigcap_{v \in U} \mathcal{LF}_v \big) \cap \big(\bigcap_{v \in U} \mathcal{LR}_v \big). \]

It is clear that  the right hand side contains only the zero vector because the normal vectors at the points of any open set on the boundary of a Euclidean sphere (or any smooth strictly convex body) span the entire space. Therefore $\rho=0$ is the only solution of H-CEQ at $p$ and, consequently, CEQ admits at most one solution for the components of the torsion tensor of a semi-symmetric linear connection point by point.

§.§ Intrinsic equations and $v$-solvability of CEQ
In this section we investigate (<ref>) evaluated at non-zero tangent vectors in $T_pM$:

\begin{equation} \label{v-CEQ}
\begin{tabu} to .5\textwidth {|ccccc|c|}
\hline
\rho_1 & \rho_2 & \rho_3 & \cdots & \rho_n & \text{RHS} \\
\hline
0 & f_{12}(v) & f_{13}(v) & \cdots & f_{1n}(v) & -\partial_1 F(v) \\
f_{21}(v) & 0 & f_{23}(v) & \cdots & f_{2n}(v) & -\partial_2 F(v) \\
f_{31} (v)& f_{32}(v) & 0 & \cdots & f_{3n}(v) & -\partial_3 F(v) \\
\vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\
f_{n1} (v)& f_{n2}(v) & f_{n3} (v)& \cdots & 0 & -\partial_n F(v) \\
\hline
\end{tabu}}
\end{equation}

The system of the compatibility equations is called $v$-solvable for $\rho$ at the point $p\in M$ if (<ref>) is solvable for any non-zero element $v\in T_pM$.

The system of the compatibility equations is $v$-solvable if and only if all vertical contact vectors are horizontal contact. It is an obvious necessary condition because the coefficient matrix of CEQ is zero at a vertically contact point and the system must be homogeneous with vanishing horizontal derivatives of the Finsler metric with respect to the compatible linear connection. The sufficiency is based on the idea of the extremal compatible linear connection [9]. The extremal solution typically depends on the reference element $v\in T_pM$  but does not take a decomposable form in general. Therefore $v$-solvability for $\rho$ needs additional conditions. Using Corollary <ref> and basic linear algebra we can formulate the following characterizations of $v$-solvability in case of semi-symmetric compatible linear connections.

The system of the compatibility equations is $v$-solvable for $\rho$ at the point $p\in M$ if and only if the following are satisfied:

* all vertically contact elements are also horizontal contact in $T_pM$ and

* the rank of the augmented matrix of system (<ref>) is 2  at all not vertically contact elements $v\in T_pM$, i.e. $H_v \in \mathrm{span}(f_1(v), \dots, f_n(v))$, where the vector $H_v$ is defined by formula (<ref>).

The system of the compatibility equations is $v$-solvable for $\rho$ at the point $p\in M$ if and only if the following are satisfied:

* all vertically contact elements are also horizontal contact  in $T_pM$  and

* for any triplets of distinct indices $i,j,k$ we have

\begin{equation} \label{solv-symm}
f_{ij}(v) \, \partial_k F (v)+ f_{jk}(v) \, \partial_i F (v)+ f_{ki}(v) \, \partial_j F(v) = 0
\end{equation}

provided that  $f_{ij}(v)\neq 0$.

If $v$ is vertically contact, (<ref>) stands trivially. Otherwise we are going to show that it is equivalent to the augmented matrix having rank 2. Suppose that $v$ is not vertically contact and choose indices $i\neq j$ such that $f_{ij}(v)\neq 0$ and $(f_i(v), f_j(v))$ is a basis of $\mathrm{span}(G_v, C_v)$. By Lemma <ref>, if $k\neq i, j$ then

\[ f_k = \dfrac{f_{kj}}{f_{ij}} \cdot f_i + \dfrac{f_{ik}}{f_{ij}} \cdot  f_j \hspace{1cm} (k=1, \dots, n). \]

In other words we can eliminate the $k$-th row for any $k\neq i, j$. The elimination must also yield zeroes on the right-hand side of (<ref>) to have a solution, i.e. for $k \neq i, j$ we must have

\[ \begin{array}{c}
-\partial_k F - \dfrac{f_{kj}}{f_{ij}} \cdot (-\partial_i F) - \dfrac{f_{ik}}{f_{ij}} \cdot (-\partial_j F) = 0 \\[12pt]
f_{ij} \, \partial_k F - f_{kj} \, \partial_i F - f_{ik} \, \partial_j F = 0.
\end{array} \]

Equation (<ref>) follows by interchanging the indices in $f_{kj}$ and $f_{ik}$. Since the coefficient matrix is of maximal rank $2$, the augmented matrix is of maximal rank $2$ as the extension of the coefficient matrix with $-\partial_i F$ and $-\partial_j F$ in the corresponding rows.

Equations (<ref>) do not contain unknown quantities. They are intrinsic conditions of the solvability. Taking $f_{ij}(v)\neq 0$ for some fixed indices $i$ and $j$ at a not vertically contact element, they provide $n-2$ equations to be automatically satisfied because $k=1, \ldots, n$, but $k\neq i, j$. The missing equations are

\begin{equation}  \label{CEQ-elim}
\left.\begin{array}{rcl}
\dotprod{f_i,\rho} & = & -\partial_i F \\[4pt]
\dotprod{f_j,\rho} & = & -\partial_j F \\
\end{array}\right\},
\end{equation}

where  $\dotprod{f_i,\rho}$ and $ \dotprod{f_j,\rho}$ stand for the inner product at $p\in M$ coming from the Riemannian metric $\gamma$. They provide the only possible solution $\rho$  in an explicit form.

§.§ The only possible solution of CEQ at the point $p\in M$
Recall that if the tangent space at $p\in M$ contains only vertically contact non-zero elements (vertically contact tangent space)  and the Finsler manifold admits a compatible (semi-symmetric) linear connection $\nabla$, then it is a Riemannian manifold because the linear isometries via the parallel transports with respect to $\nabla$ extend the quadratic Finslerian (esp. Riemannian) indicatrix at the point $p$ to the entire (connected) manifold. Therefore we present the solution of CEQ in the generic case of non-Riemannian Finsler manifolds. Let us choose a not vertically contact element $v \in T_p^{\circ} M$ and indices $i\neq j$ with $f_{ij}(v)\neq 0$, i.e. $f_i, f_j$ are linearly independent over some neighborhood $U$ of $v$ in $T_pM$. Using (<ref>), the eliminated form (<ref>) of CEQ gives that

\[  {\arraycolsep=1pt \left.\begin{array}{ccccl}
y^i \dotprod{G,\rho} & - & \dot{\partial}_i F \dotprod{C,\rho} & = & -\partial_i F \\[5pt]
y^j \dotprod{G,\rho} & - & \dot{\partial}_j F \dotprod{C,\rho} & = & -\partial_j F \\
\end{array}\right\} } \Longleftrightarrow
\begin{bmatrix} y^i & -\dot{\partial}_i F \\[5pt] y^j & -\dot{\partial}_j F  \end{bmatrix} \cdot \begin{bmatrix} \dotprod{G,\rho} \\[5pt] \dotprod{C,\rho} \end{bmatrix} = \begin{bmatrix}
-\partial_i F \\[5pt] -\partial_j F
\end{bmatrix}, \]

and, consequently,

\[ \begin{bmatrix} \dotprod{G,\rho} \\[5pt] \dotprod{C,\rho} \end{bmatrix} = \dfrac{1}{f_{ji}}\begin{bmatrix}
-\dot{\partial}_j F &\dot{\partial}_i F \\[5pt] -y^j & y^i
\end{bmatrix} \cdot \begin{bmatrix}
-\partial_i F \\[5pt] -\partial_j F
\end{bmatrix}. \]

We are going to concentrate on the second row

\begin{equation} \label{eq-c}
\dotprod{C,\rho} = \dfrac{1}{f_{ji}} \left( y^j \partial_i F - y^i \partial_j F \right) =: \dfrac{f_{ji}^h}{f_{ji}}
\end{equation}

at the points of the open neighborhood $U$ around $v$. Let us choose a value $\varepsilon>0$ such that all the elements

\[ \begin{array}{rcccl}
w_1 & := & v-\varepsilon \cdot \partial/\partial u^1(p) & = & [v^1-\varepsilon, v^2, v^3, \dots, v^n] \\
w_2 & := & v-\varepsilon \cdot \partial/\partial u^2(p) &= & [v^1, v^2-\varepsilon, v^3, \dots, v^n] \\
& \vdots &&& \\
w_n & := & v-\varepsilon \cdot \partial/\partial u^n(p) & = & [v^1, v^2, v^3, \dots, v^n-\varepsilon]
\end{array} \]

are contained in $U$. Then (<ref>) implies the system

\begin{equation} \label{ceq-indep} \begin{bmatrix}
v^1-\varepsilon & v^2 & v^3 & \cdots & v^n \\[4pt]
v^1 & v^2-\varepsilon & v^3 & \dots & v^n \\[4pt]
\vdots & \vdots & \vdots & \ddots & \vdots \\[4pt]
v^1 & v^2 & v^3 & \dots & v^n-\varepsilon
\end{bmatrix} \cdot \begin{bmatrix} \rho_1 \\[4pt] \rho_2 \\[4pt] \vdots \\[4pt] \rho_n \end{bmatrix} = \begin{bmatrix}
f^h_{ji}/f_{ji}(w_1) \\[4pt] f^h_{ji}/f_{ji}(w_2) \\[4pt] \vdots \\[4pt] f^h_{ji}/f_{ji}(w_n) \end{bmatrix}
\end{equation}

of linear equations. Using the notation

\[ V:= \begin{bmatrix}
v^1 & v^2 & \cdots & v^n \\
\vdots & \vdots & \ddots & \vdots \\
v^1 & v^2 & \cdots & v^n
\end{bmatrix}, \]

we have to investigate the regularity of the matrix $V-\varepsilon I$, where $I$ denotes the identity matrix of the same type as $V$.

The matrix $V-\varepsilon I$ is regular if and only if $\varepsilon \notin \{ 0, \widetilde{v}:=v^1+\dots+v^n \}$.

The determinant $\mathrm{det}(V-\varepsilon I)$ is the characteristic polynomial of $V$ with $\varepsilon$ as the variable. It is zero if and only if $\varepsilon$ is an eigenvalue of $V$. Consider the transpose of $V$ as the matrix of a linear transformation $\varphi$ (the eigenvalues are the same as those of $V$). Since the image of $\varphi$ is the line generated by $v$, it follows that $\varphi$ has a kernel of dimension $n-1$ and $v$ is an eigenvector corresponding to the eigenvalue $\widetilde{v}:=v^1+\dots+v^n$ because of

\[ \varphi(v)= \begin{bmatrix}
v^1 & \cdots & v^1 \\ \vdots & \ddots & \vdots \\ v^n & \cdots & v^n \end{bmatrix} \cdot \begin{bmatrix} v^1 \\ \vdots \\ v^n\end{bmatrix} = \begin{bmatrix}
v^1 (v^1+\dots+v^n) \\ \vdots \\v^n(v^1+\dots+v^n)
\end{bmatrix} = \widetilde{v} \cdot v. \qedhere\]

Choosing $\varepsilon \notin \{ 0, \widetilde{v}:=v^1+\dots+v^n\}$, we have

\[ \left[ V-\varepsilon I \right]^{-1} = \dfrac{1}{(\widetilde{v}-\varepsilon)\varepsilon} \left[ V-(\widetilde{v}-\varepsilon) I \right]. \]

We shall prove the formula

\[ \left[V-\varepsilon I\right] \cdot \left[ V-(\widetilde{v}-\varepsilon)I\right] = (\widetilde{v}-\varepsilon)\varepsilon I.\]

Rearranging the left-hand side,

\[ V \cdot V - (\widetilde{v}-\varepsilon)V-\varepsilon V + \varepsilon (\widetilde{v}-\varepsilon)I=(\widetilde{v}-\varepsilon)\varepsilon I \]

because of $V\cdot V=\widetilde{v} \cdot V$.

Returning to (<ref>),

\[\begin{bmatrix} \rho_1 \\[4pt] \rho_2 \\[4pt] \vdots \\[4pt] \rho_n \end{bmatrix} =  \dfrac{1}{(\widetilde{v}-\varepsilon)\varepsilon} \begin{bmatrix}
v^1+\varepsilon-\widetilde{v} & v^2 &  \cdots & v^n \\[4pt]
v^1 & v^2+\varepsilon-\widetilde{v} &  \dots & v^n \\[4pt]
\vdots & \vdots &  \ddots & \vdots \\[4pt]
v^1 & v^2 &  \dots & v^n+\varepsilon-\widetilde{v}
\end{bmatrix} \cdot \begin{bmatrix}
f^h_{ji}/f_{ji}(w_1) \\[4pt] f^h_{ji}/f_{ji}(w_2) \\[4pt] \vdots \\[4pt] f^h_{ji}/f_{ji}(w_n) \end{bmatrix}. \]

If a non-Riemannian Finsler manifold admits a semi-symmetric compatible linear connection, then the only possible values of the components $\rho_k$ in formula  (<ref>) for its torsion at the point $p\in M$ are

\begin{equation}
\rho_k = \dfrac{1}{\varepsilon} \left( \dfrac{1}{\widetilde{v}-\varepsilon} \sum_{l=1}^n v^l \dfrac{f^h_{ji}}{f_{ji}}(w_l) - \dfrac{f^h_{ji}}{f_{ji}}(w_k) \right),
\end{equation}

* $v=[v^1, \dots, v^n]$ is a not vertically contact vector in $T^{\circ}_p M$,

* $\varepsilon \notin \{ 0, \widetilde{v}:=v^1+\dots+v^n\}$ is a radius of a closed ball around $v$ in $T_pM$ all of whose elements are also not vertically contact,

* $w_i=[v^1, \dots, v^{i-1}, v^i-\varepsilon, v^{i+1}, \dots, v^n]$, $i=1, \ldots, n$,

* $f^h_{ji}= y^j \partial_i F - y^i \partial_j F = y^j \dfrac{\partial F}{\partial x^i} - y^i \dfrac{\partial F}{\partial x^j}$,

* $f_{ji}= y^j \dot{\partial}_i F - y^i \dot{\partial}_j F = y^j \dfrac{\partial F}{\partial y^i} - y^i \dfrac{\partial F}{\partial y^j}$ and

* the coordinates on the base manifold form a normal coordinate system with respect to the averaged Riemannian metric $\gamma$ around the point $p\in M$.

§ ACKNOWLEDGMENTS

Márk Oláh is supported by the UNKP-21-3 New National Excellence Program of the Ministry for Innovation and Technology from the source of the National Research, Development and Innovation Fund of Hungary.

[1] N. Bartelmeß  and V. Matveev, Monochromatic metrics are generalized Berwald, J. Diff. Geom. Appl. 58 (2018), pp. 264-271.

[2] M. Crampin, On the construction of Riemannian metrics for Berwald spaces by averaging, Houston J. Math. 40 (3) (2014), pp. 737–750.

[3] V. S. Matveev and M. Troyanov, The Binet-Legendre metric in Finsler geometry, Geometry and Topology 16 (2012), pp. 2135-2170.

[4] D. Bao, S.-S. Chern and Z. Shen, An Introduction to Riemann-Finsler geometry, Springer-Verlag, 2000.

[5] Cs. Vincze, A new proof of Szabó' s theorem on the Riemann-metrizability of Berwald manifolds, J. AMAPN 21 (2005), pp. 199-204.

[6] Cs. Vincze, Generalized Berwald manifolds with semi-symmetric linear connections, Publ. Math. Debrecen 83 (4) (2013), pp. 741-755.

[7] Cs. Vincze, On a special type of generalized Berwald manifolds: semi-symmetric linear connections preserving the Finslerian length of tangent vectors, "Finsler geometry: New methods and Perspectives", European Journal of Mathematics 3 (4) (2017), pp. 1098 - 1171.

[8] Cs. Vincze, M. Oláh, L. Muhsin, On the divergence representation of the Gauss curvature of Riemannian surfaces and its applications, Rend. Circ. Mat. Palermo, II Ser. 69 (2020), pp. 1-13. https:// doi.org/10.1007/s12215-018-0382-6.

[9] Cs. Vincze, On the extremal compatible linear connection of a generalized Berwald manifold, Aequat. Math. 96 (2022), pp. 53–70. https://doi.org/10.1007/s00010-021-00859-x. ArXiv:1909.03096.

[10] Cs. Vincze, M. Oláh, On the extremal compatible linear connection of a Randers space, J. Geom. 111 (19) (2020). https://doi.org/10.1007/s00022-020-00532-9.

[11] V. Wagner, On generalized Berwald spaces, CR Dokl. Acad. Sci. USSR (N.S.) 39 (1943), pp. 3-5.


# Arithmetic properties for generalized cubic partitions and overpartitions
modulo a prime

Tewodros Amdeberhan  Department of Mathematics, Tulane University, New
Orleans, LA 70118, USA<EMAIL_ADDRESS>,  James A. Sellers  Department of
Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812,
USA<EMAIL_ADDRESS>and  Ajit Singh  Department of Mathematics, University
of Virginia, Charlottesville, VA 22904, USA<EMAIL_ADDRESS>
(Date: August 28, 2024)

###### Abstract.

A cubic partition is an integer partition wherein the even parts can appear in
two colors. In this paper, we introduce the notion of generalized cubic
partitions and prove a number of new congruences akin to the classical
Ramanujan-type. We emphasize two methods of proofs, one elementary (relying
significantly on functional equations) and the other based on modular forms.
We close by proving analogous results for generalized overcubic partitions.

###### Key words and phrases:

cubic partitions, Ramanujan congruences, modular forms

###### 2020 Mathematics Subject Classification:

05A17, 11F03, 11P83

##  1\. Introduction

A partition $\lambda$ of a positive integer $n$ is a sequence of positive
integers $\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{r}$ such that
$\sum_{i=1}^{r}\lambda_{i}=n$. The values
$\lambda_{1},\lambda_{2},\dots,\lambda_{r}$ are called the parts of $\lambda$.
We let $p(n)$ denote the number of partitions of $n$ for $n\geq 1$, and we
define $p(0):=1.$ As an example, note that the partitions of $n=4$ are

$4,\ \ 3+1,\ \ 2+2,\ \ 2+1+1,\ \ 1+1+1+1$

and this means $p(4)=5.$

Throughout this work, we adopt the notations $(a;q)_{\infty}=\prod_{j\geq
0}(1-aq^{j})$, where $\lvert q\rvert<1$, and $f_{k}:=(q^{k};q^{k})_{\infty}$.
As was proven by Euler, we know that

$\displaystyle P(q):=\sum_{n\geq 0}p(n)\,q^{n}$ $\displaystyle=\prod_{j\geq
1}\frac{1}{1-q^{j}}=\frac{1}{f_{1}}.$

Because the focus of this paper is on divisibility properties of certain
partition functions, we remind the reader of the celebrated Ramanujan
congruences for $p(n)$ [10, p. 210, p. 230]: For all $n\geq 0,$

$\displaystyle p(5n+4)$ $\displaystyle\equiv 0\pmod{5},$ (1.1) $\displaystyle
p(7n+5)$ $\displaystyle\equiv 0\pmod{7},\textrm{\ \ and}$ $\displaystyle
p(11n+6)$ $\displaystyle\equiv 0\pmod{11}.$

Indeed, among Ramanujan’s discoveries, the equality

(1.2) $\sum_{n\geq
0}p(5n+4)\,q^{n}=5\,\frac{(q^{5};q^{5})_{\infty}^{5}}{(q;q)_{\infty}^{6}}$

was regarded as his “Most Beautiful Identity” by both Hardy and MacMahon (see
[10, p. xxxv]).

Based on an identity on Ramanujan’s cubic continued fractions [1], Chan [3]
introduced the notion of cubic partitions. A _cubic partition_ of weight $n$
is a partition of $n$ wherein the even parts can appear in two colors. We
denote the number of such cubic partitions by $a_{2}(n)$ and define
$a_{2}(0):=1$. For example, $a_{2}(3)=4$ since the list of cubic partitions of
$n=3$ is: $3,\
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+1,\ 2+1,\
1+1+1$. We also compute $a_{2}(4)=9$ because the cubic partitions of $4$ are

${\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}4},\ \ 4,\
\ 3+1,\ \
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{2}}+{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}{2}},\
\ {\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+2,\ \
2+2,\ \
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+1+1,\ \
2+1+1,\ \ 1+1+1+1.$

It is clear from the definition of cubic partitions that the generating
function for $a_{2}(n)$ is given by

$\displaystyle F_{2}(q):=\sum_{n\geq 0}a_{2}(n)\,q^{n}$
$\displaystyle=\prod_{j\geq
1}\frac{1}{(1-q^{j})(1-q^{2j})}=\frac{1}{f_{1}f_{2}}.$

Chan [3] succeeded in proving the following elegant analogue of (1.2):

$\sum_{n\geq
0}a_{2}(3n+2)\,q^{n}=3\,\frac{(q^{3};q^{3})_{\infty}^{3}(q^{6};q^{6})_{\infty}^{3}}{(q;q)_{\infty}^{4}(q^{2};q^{2})_{\infty}^{4}}.$

It is immediate that, for all $n\geq 0,$

(1.3) $a_{2}(3n+2)\equiv 0\pmod{3},$

a result similar in nature to (1.1). Since then, many authors have studied
similar congruences for $a_{2}(n)$ (see [5], [6] and references therein). In
particular, we highlight the following result of Chan and Toh [4] which was
proven using modular forms:

######  Theorem 1.1.

For all $n\geq 0,$

$a_{2}(5^{j}n+d_{j})\equiv 0\pmod{5^{\lfloor j/2\rfloor}}$

where $d_{j}$ is the inverse of 8 modulo $5^{j}$.

By way of generalization, we define a _generalized cubic partition_ of weight
$n$ to be a partition of $n\geq 1$ wherein each even part may appear in $c\geq
1$ different colors. We denote the number of such generalized cubic partitions
by $a_{c}(n)$ and define $a_{c}(0):=1$. Notice that, for all $n\geq 0$,
$a_{1}(n)=p(n)$ while $a_{2}(n)$ enumerates the cubic partitions of $n$ as
described above. It is clear that the generating function for $a_{c}(n)$ is
given by

(1.4) $\displaystyle F_{c}(q):=\sum_{n\geq 0}a_{c}(n)\,q^{n}$
$\displaystyle=\prod_{j\geq
1}\frac{1}{(1-q^{j})(1-q^{2j})^{c-1}}=\frac{1}{f_{1}f_{2}^{c-1}}.$

Our primary goal in this work is to prove several congruences modulo primes
satisfied by $a_{c}(n)$ for an infinite family of values of $c$. The proof
techniques that we will utilize will include elementary approaches as well as
modular forms. In particular, we will prove the following infinite family of
congruences modulo a prime $p\geq 3$ which are reminiscent of (1.3):

######  Theorem 1.2.

Let $p$ be an odd prime. Then, for all $n\geq 0$,

$a_{p-1}(pn+r)\equiv 0\pmod{p}$

where $r$ is an integer, $1\leq r\leq p-1$, such that $8r+1$ is a quadratic
nonresidue modulo $p$.

We will then show that Theorem 1.2 can be generalized in the following natural
way:

######  Corollary 1.3.

Let $p\geq 3$ be prime and let $r$, $1\leq r\leq p-1$ such that, for all
$n\geq 0,$ $a_{p-1}(pn+r)\equiv 0\pmod{p}$. Then, for any $k\geq 1$, and for
all $n\geq 0,$ $a_{kp-1}(pn+r)\equiv 0\pmod{p}$.

We will also prove two “isolated” congruences, modulo the primes 7 and 11,
respectively.

######  Theorem 1.4.

For all $n\geq 0,$

$\displaystyle a_{3}(7n+4)$ $\displaystyle\equiv 0\pmod{7}\textrm{\ \ and}$
$\displaystyle a_{5}(11n+10)$ $\displaystyle\equiv 0\pmod{11}.$

These two results will be proved via modular forms (using arguments that are
very similar to one another).

The remainder of this paper is organized as follows. In Section 2 we present
elementary proofs of Theorem 1.2 and Corollary 1.3. Section 3 contains the
basic results from the theory of modular forms which we need to prove Theorem
1.4. We close that section with the proof of Theorem 1.4. Finally, in Section
4, we consider _generalized cubic overpartitions_ and prove an infinite family
of congruences for the related partition functions via elementary methods. We
conclude Section 4 with some relevant remarks.

##  2\. Elementary Proofs of Theorem 1.2 and Corollary 1.3

Sellers [11, Theorem 2.1] developed, among several other results, a functional
equation for $F_{2}(q)$. That result can be generalized to our family of
generating functions. Namely, thanks to (1.4), we can easily show that

(2.1) $F_{c}(q)=\psi(q)\psi(q^{2})^{c-1}F_{c}(q^{2})^{2}$

where

$\psi(q):=\frac{f_{2}^{2}}{f_{1}}=\sum_{k\geq 0}q^{k(k+1)/2}$

is one of Ramanujan’s well–known theta functions [2, p. 6]. We can then
iterate (2.1) to prove the following:

######  Lemma 2.1.

Let $p\geq 3$ be prime. Then

$F_{p-1}(q)=\psi(q)\prod_{i\geq 1}\psi(q^{2^{i}})^{p\cdot 2^{i-1}}.$

###### Proof.

We have

$\displaystyle F_{p-1}(q)$
$\displaystyle=\frac{1}{f_{1}f_{2}^{p-2}}=\psi(q)\psi(q^{2})^{p-2}F_{p-1}(q^{2})^{2}$
$\displaystyle=\psi(q)\psi(q^{2})^{p-2}\left(\psi(q^{2})\psi(q^{4})^{p-2}F_{p-1}(q^{4})^{2}\right)^{2}$
$\displaystyle=\psi(q)\psi(q^{2})^{p}\psi(q^{4})^{2(p-2)}F_{p-1}(q^{4})^{4}$
$\displaystyle=\psi(q)\psi(q^{2})^{p}\psi(q^{4})^{2(p-2)}\left(\psi(q^{4})\psi(q^{8})^{p-2}F_{p-1}(q^{8})^{2}\right)^{4}$
$\displaystyle=\psi(q)\psi(q^{2})^{p}\psi(q^{4})^{2p}\psi(q^{8})^{4(p-2)}F_{p-1}(q^{8})^{8}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\vdots$

The result follows by continuing to iterate (2.1) indefinitely. ∎

Lemma 2.1 then leads to a straightforward proof of Theorem 1.2.

###### Proof of Theorem 1.2.

Thanks to Lemma 2.1, we know

$F_{p-1}(q)=\psi(q)\prod_{i\geq 1}\psi(q^{2^{i}})^{p\cdot
2^{i-1}}\equiv\psi(q)\prod_{i\geq 1}\psi(q^{p\cdot 2^{i}})^{2^{i-1}}\pmod{p}.$

Thus, if we wish to focus our attention on $a_{p-1}(pn+r)$ with the conditions
as stated in the theorem, then we only need to consider $\psi(q)$ because

$\prod_{i\geq 1}\psi(q^{p\cdot 2^{i}})^{2^{i-1}}$

is a function of $q^{p}.$ Thus, we simply need to confirm that we have no
solutions to the equation

$pn+r=\frac{k(k+1)}{2}$

or

$8(pn+r)+1=8\left(\frac{k(k+1)}{2}\right)+1=(2k+1)^{2}.$

If we did have a solution to the above, then

$8r+1\equiv(2k+1)^{2}\pmod{p},$

but we have explicitly assumed that $r$ has been selected so that $8r+1$ is a
quadratic nonresidue modulo $p$. Therefore, there can be no such solutions,
and this proves our result. ∎

###### Proof of Corollary 1.3.

The generating function for $a_{kp-1}(n)$ is

$\displaystyle\frac{1}{f_{1}f_{2}^{kp-2}}$
$\displaystyle=\frac{1}{f_{1}f_{2}^{p-2}f_{2}^{(k-1)p}}\equiv\frac{1}{f_{1}f_{2}^{p-2}f_{2p}^{k-1}}\pmod{p}$

If we wish to focus our attention on $a_{kp-1}(pn+r)$ with the conditions as
stated in the corollary, then we only need to look at
$\frac{1}{f_{1}f_{2}^{p-2}}$ because $f_{2p}^{k-1}$ is a function of $q^{p}$.
Notice that $\frac{1}{f_{1}f_{2}^{p-2}}$ is the generating function for
$a_{p-1}(n).$ Given that a congruence mod $p$ is assumed to hold for
$a_{p-1}(n)$, it must be the case that a similar congruence modulo $p$ will
hold for $a_{kp-1}(n)$ on the same arithmetic progression. ∎

##  3\. A Modular Forms Proof of Theorem 1.4

We begin this section with some definitions and basic facts on modular forms
that are instrumental in furnishing our proof of Theorem 1.4. For additional
details, see for example [8, 9]. We first identify the matrix groups

$\displaystyle\text{SL}_{2}(\mathbb{Z})$
$\displaystyle:=\left\\{\begin{bmatrix}a&b\\\
c&d\end{bmatrix}:a,b,c,d\in\mathbb{Z},ad-bc=1\right\\},$
$\displaystyle\Gamma_{0}(N)$ $\displaystyle:=\left\\{\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\text{SL}_{2}(\mathbb{Z}):c\equiv 0\pmod{N}\right\\},$
$\displaystyle\Gamma_{1}(N)$ $\displaystyle:=\left\\{\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\Gamma_{0}(N):a\equiv d\equiv 1\pmod{N}\right\\},$

and

$\displaystyle\Gamma(N)$ $\displaystyle:=\left\\{\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\text{SL}_{2}(\mathbb{Z}):a\equiv d\equiv
1\pmod{N},\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ b\equiv c\equiv
0\pmod{N}\right\\},$

where $N$ is a positive integer. A subgroup $\Gamma$ of the group
$\text{SL}_{2}(\mathbb{Z})$ is called a _congruence subgroup_ if
$\Gamma(N)\subseteq\Gamma$ for some $N$. The smallest $N$ such that
$\Gamma(N)\subseteq\Gamma$ is called the _level of $\Gamma$_. For example,
$\Gamma_{0}(N)$ and $\Gamma_{1}(N)$ are congruence subgroups of level $N$.

Let $\mathbb{H}:=\\{z\in\mathbb{C}:\text{Im}(z)>0\\}$ be the upper half of the
complex plane. Then, the following subgroup of the general linear group

$\text{GL}_{2}^{+}(\mathbb{R})=\left\\{\begin{bmatrix}a&b\\\
c&d\end{bmatrix}:a,b,c,d\in\mathbb{R}\leavevmode\nobreak\
\text{and}\leavevmode\nobreak\ ad-bc>0\right\\}$

acts on $\mathbb{H}$ by $\begin{bmatrix}a&b\\\
c&d\end{bmatrix}z=\displaystyle\frac{az+b}{cz+d}$. We identify $\infty$ with
$\displaystyle\frac{1}{0}$ and define

$\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\displaystyle\frac{r}{s}=\displaystyle\frac{ar+bs}{cr+ds},$

where $\displaystyle\frac{r}{s}\in\mathbb{Q}\cup\\{\infty\\}$. This gives an
action of $\text{GL}_{2}^{+}(\mathbb{R})$ on the extended upper half-plane
$\mathbb{H}^{\ast}=\mathbb{H}\cup\mathbb{Q}\cup\\{\infty\\}$. Suppose that
$\Gamma$ is a congruence subgroup of $\text{SL}_{2}(\mathbb{Z})$. A _cusp_ of
$\Gamma$ is an equivalence class in
$\mathbb{P}^{1}=\mathbb{Q}\cup\\{\infty\\}$ under the action of $\Gamma$. The
group $\text{GL}_{2}^{+}(\mathbb{R})$ also acts on functions
$f:\mathbb{H}\rightarrow\mathbb{C}$. In particular, suppose that
$\gamma=\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\text{GL}_{2}^{+}(\mathbb{R})$. If $f(z)$ is a meromorphic
function on $\mathbb{H}$ and $\ell$ is an integer, then define the slash
operator $|_{\ell}$ by

$(f|_{\ell}\gamma)(z):=(\text{det}\leavevmode\nobreak\
{\gamma})^{\ell/2}(cz+d)^{-\ell}f(\gamma z).$

######  Definition 3.1.

Let $\Gamma$ be a congruence subgroup of level $N$. A holomorphic function
$f:\mathbb{H}\rightarrow\mathbb{C}$ is called a _modular form_ with integer
weight $\ell$ on $\Gamma$ if the following hold:

  1. (1)

We have $f\left(\displaystyle\gamma z\right)=(cz+d)^{\ell}f(z)$ for all
$z\in\mathbb{H}$ and all $\gamma=\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\Gamma$.

  2. (2)

If $\gamma\in\text{SL}_{2}(\mathbb{Z})$, then $(f|_{\ell}\gamma)(z)$ has a
Fourier expansion of the form

$(f|_{\ell}\gamma)(z)=\displaystyle\sum_{n\geq 0}a_{\gamma}(n)q_{N}^{n},$

where $q_{N}:=e^{2\pi iz/N}$.

For a positive integer $\ell$, the complex vector space of modular forms of
weight $\ell$ with respect to a congruence subgroup $\Gamma$ is denoted by
$M_{\ell}(\Gamma)$.

######  Definition 3.2.

[9, Definition 1.15] If $\chi$ is a Dirichlet character modulo $N$, then we
say that a modular form $f\in M_{\ell}(\Gamma_{1}(N))$ has _Nebentypus
character_ $\chi$ if

$f\left(\frac{az+b}{cz+d}\right)=\chi(d)(cz+d)^{\ell}f(z)$

for all $z\in\mathbb{H}$ and all $\begin{bmatrix}a&b\\\
c&d\end{bmatrix}\in\Gamma_{0}(N)$. The space of such modular forms is denoted
by $M_{\ell}(\Gamma_{0}(N),\chi)$.

In this paper, the relevant modular forms are those that arise from eta-
quotients. The _Dedekind eta-function_ $\eta(z)$ is defined by

$\displaystyle\eta(z):=q^{1/24}(q;q)_{\infty}=q^{1/24}\prod_{n=1}^{\infty}(1-q^{n}),$

where $q:=e^{2\pi iz}$ and $z\in\mathbb{H}$, the upper half-plane. A function
$f(z)$ is called an _eta-quotient_ if it is of the form

$\displaystyle f(z)=\prod_{\delta\mid N}\eta(\delta z)^{r_{\delta}},$

where $N$ is a positive integer and $r_{\delta}$ is an integer. We now recall
two valuable theorems from [9, p. 18] which will be used to prove our results.

######  Theorem 3.3.

[9, Theorem 1.64] If $f(z)=\prod_{\delta\mid N}\eta(\delta z)^{r_{\delta}}$ is
an eta-quotient such that $\ell=\frac{1}{2}\sum_{\delta\mid
N}r_{\delta}\in\mathbb{Z}$,

$\sum_{\delta\mid N}\delta r_{\delta}\equiv 0\pmod{24}\ \ \ \ \ \ \text{and}\
\ \ \ \ \ \sum_{\delta\mid N}\frac{N}{\delta}r_{\delta}\equiv 0\pmod{24},$

then $f(z)$ satisfies $f\left(\gamma z\right)=\chi(d)(cz+d)^{\ell}f(z)$ for
every $\gamma=\begin{bmatrix}a&b\\\ c&d\end{bmatrix}\in\Gamma_{0}(N)$. Here
the character $\chi$ is defined by
$\chi(\bullet):=\left(\frac{(-1)^{\ell}s}{\bullet}\right)$, where
$s:=\prod_{\delta\mid N}\delta^{r_{\delta}}$.

Suppose that $f$ is an eta-quotient satisfying the conditions of Theorem 3.3
and that the associated weight $\ell$ is a positive integer. If the function
$f(z)$ is holomorphic at all of the cusps of $\Gamma_{0}(N)$, then $f(z)\in
M_{\ell}(\Gamma_{0}(N),\chi)$. The next theorem gives a necessary condition
for determining orders of an eta-quotient at cusps.

######  Theorem 3.4.

[9, Theorem 1.65] Let $c,d$ and $N$ be positive integers with $d\mid N$ and
$\gcd(c,d)=1$. If $f$ is an eta-quotient satisfying the conditions of Theorem
3.3 for $N$, then the order of vanishing of $f(z)$ at the cusp $\frac{c}{d}$
is

$\frac{N}{24}\sum_{\delta\mid
N}\frac{\gcd(d,\delta)^{2}r_{\delta}}{\gcd(d,\frac{N}{d})d\delta}.$

We now remind ourselves a result of Sturm [12] which gives a criterion to test
whether two modular forms are congruent modulo a given prime.

######  Theorem 3.5.

Let $k$ be an integer and $g(z)=\sum_{n=0}^{\infty}a(n)q^{n}$ a modular form
of weight $k$ for $\Gamma_{0}(N)$. For any given positive integer $m$, if
$a(n)\equiv 0\pmod{m}$ holds for all
$\displaystyle{n\leq\frac{kN}{12}\prod_{p\leavevmode\nobreak\
\text{prime}\atop p\,|\,N}\leavevmode\nobreak\ \left(1+\frac{1}{p}\right)}\,,$
then $a(n)\equiv 0\pmod{m}$ holds for any $n\geq 0$.

We now recall the definition of Hecke operators. Let $m$ be a positive integer
and $f(z)=\sum_{n=0}^{\infty}a(n)q^{n}\in M_{\ell}(\Gamma_{0}(N),\chi)$. Then
the action of Hecke operator $T_{m}$ on $f(z)$ is defined by

$\displaystyle
f(z)\,|\,T_{m}:=\sum_{n=0}^{\infty}\left(\sum_{d\mid\gcd(n,m)}\chi(d)d^{\ell-1}a\left(\frac{nm}{d^{2}}\right)\right)q^{n}.$

In particular, if $m=p$ is prime, we have

(3.1) $\displaystyle
f(z)\,|\,T_{p}:=\sum_{n=0}^{\infty}\left(a(pn)+\chi(p)p^{\ell-1}a\left(\frac{n}{p}\right)\right)q^{n}.$

We take by convention that $a(n/p)=0$ whenever $p\nmid n$. The next result
follows directly from (3.1).

######  Proposition 3.6.

Let $p$ be a prime, $g(z)\in{\mathbb{Z}}[[q]],h(z)\in{\mathbb{Z}}[[q^{p}]]$,
and $k>1$. Then, we have

$\displaystyle\left(g(z)h(z)\right)\,|\,T_{p}\equiv\left(g(z)|T_{p}\cdot
h(z/p)\right)\pmod{p}.$

With the above in place, we are ready to supply the proof of Theorem 1.4.

###### Proof of Theorem 1.4.

We begin by proving the mod 7 congruence that appears in the theorem. By
taking $c=3$ in (1.4), we have

(3.2) $\displaystyle\sum_{n\geq 0}a_{3}(n)\,q^{n}$ $\displaystyle=\prod_{k\geq
1}\frac{1}{(1-q^{k})(1-q^{2k})^{2}}.$

Let

$H(z):=\frac{\eta^{76}(z)}{\eta^{2}(2z)}.$

By Theorems 3.3 and 3.4, we find that $H(z)$ is a modular form of weight $37$,
level $8$ and character $\chi_{1}=(\frac{-2^{-2}}{\bullet})$. By (3.2), the
Fourier expansion of our form satisfies

$\displaystyle
H(z)=\left(\sum_{n=0}^{\infty}a_{3}(n)q^{n+3}\right)\prod_{k\geq
1}(1-q^{k})^{77}.$

Using Proposition 3.6, we calculate that

$\displaystyle
H(z)\,|\,T_{7}\equiv\left(\sum_{n=0}^{\infty}a_{3}(7n+4)q^{n+1}\right)\prod_{k\geq
1}(1-q^{k})^{11}\pmod{7}.$

Since the Hecke operator is an endomorphism on
$M_{37}\left(\Gamma_{0}(8),\chi_{1}\right)$, we have that $H(z)\,|\,T_{7}\in
M_{37}\left(\Gamma_{0}(8),\chi_{1}\right)$. By Theorem 3.5, the Sturm bound
for this space of forms is $37$. Using $SageMath$, we verify that the Fourier
coefficients of $H(z)\,|\,T_{7}$ up to the desired bound are congruent to 0
modulo 7. Hence, Theorem 3.5 confirms that $H(z)\,|\,T_{7}\equiv 0\pmod{7}$.
This completes the proof of the mod 7 congruence.

We next consider the mod 11 congruence in the theorem whose proof is rather
similar to the proof of the mod 7 congruence above. By (1.4), we have

(3.3) $\displaystyle\sum_{n\geq 0}a_{5}(n)\,q^{n}$ $\displaystyle=\prod_{k\geq
1}\frac{1}{(1-q^{k})(1-q^{2k})^{4}}.$

Let

$G(z):=\frac{\eta^{32}(z)}{\eta^{4}(2z)}.$

By Theorems 3.3 and 3.4, we find that $G(z)$ is a modular form of weight $14$,
level $4$ and character $\chi_{0}=(\frac{2^{-4}}{\bullet})$. By (3.3), the
Fourier expansion of our form satisfies

$\displaystyle
G(z)=\left(\sum_{n=0}^{\infty}a_{5}(n)q^{n+1}\right)\prod_{k\geq
1}(1-q^{k})^{33}.$

Using Proposition 3.6, we calculate that

$\displaystyle
G(z)\,|\,T_{11}\equiv\left(\sum_{n=0}^{\infty}a_{5}(11n+10)q^{n+1}\right)\prod_{k\geq
1}(1-q^{k})^{3}\pmod{11}.$

Since the Hecke operator is an endomorphism on
$M_{14}\left(\Gamma_{0}(4),\chi_{0}\right)$, we have that $G(z)\,|\,T_{11}\in
M_{14}\left(\Gamma_{0}(4),\chi_{0}\right)$. By Theorem 3.5, the Sturm bound
for this space of forms is $7$. Hence, Theorem 3.5 confirms that
$G(z)\,|\,T_{11}\equiv 0\pmod{11}$. This completes the proof of the mod 11
congruence. ∎

##  4\. Closing Thoughts

We now transition to a consideration of combinatorial objects related to
generalized cubic partitions, namely, generalized cubic overpartitions.

An _overpartition_ of $n$ is a partition of $n$ in which the first occurrence
of a part may be overlined. In 2010, Kim [7] introduced an overpartition
version of cubic partitions, sometimes called _overcubic partitions_.
Following the four cubic partitions of $n=3$ from the Introduction, we find
that the corresponding $12$ overcubic partitions of $n=3$ are given by the
following:

$\displaystyle 3,\ \ \overline{3},\ \ 2+1,\ \
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+1,\ \
\overline{2}+1,\ \
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\overline{2}}+1,$
$\displaystyle 2+\overline{1},\ \
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}2}+\overline{1},\
\ \overline{2}+\overline{1},\ \
{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\overline{2}}+\overline{1},\
\ 1+1+1,\ \ \overline{1}+1+1.$

If we now generalize these to mirror our generalization of cubic partitions,
then the generating function for these cubic overpartitions is given by

$\overline{F}_{c}(q):=\sum_{n\geq
0}\overline{a}_{c}(n)\,q^{n}=\frac{(-q;q)_{\infty}(-q^{2};q^{2})^{c-1}_{\infty}}{(q;q)_{\infty}(q^{2};q^{2})_{\infty}^{c-1}}.$

We now prove the following natural companion to Corollary 1.3.

######  Theorem 4.1.

Let $p\geq 3$ be prime, and let $r$, $1\leq r\leq p-1$, such that r is a
quadratic nonresidue modulo $p$. Also let $k$ be a positive integer. Then, for
all $n\geq 0,$ $\overline{a}_{kp-1}(pn+r)\equiv 0\pmod{p}.$

###### Proof.

After elementary manipulations, the generating function for
$\overline{a}_{c}(n)$ can be given in the form

$\overline{F}_{c}(q)=\frac{f_{4}^{c-1}}{f_{1}^{2}f_{2}^{2c-3}}.$

Now consider the above generating function for $c=kp-1$. Then we have

$\displaystyle\sum_{n\geq 0}\overline{a}_{kp-1}(n)q^{n}$
$\displaystyle=\frac{f_{4}^{kp-2}}{f_{1}^{2}f_{2}^{2kp-5}}=\left(\frac{f_{4}^{kp}}{f_{2}^{2kp}}\right)\left(\frac{f_{2}^{5}}{f_{1}^{2}f_{4}^{2}}\right)=\left(\frac{f_{4}^{kp}}{f_{2}^{2kp}}\right)\cdot\varphi(q)$
$\displaystyle\equiv\left(\frac{f_{4p}^{k}}{f_{2p}^{2k}}\right)\cdot\varphi(q)\pmod{p}$

where

$\varphi(q):=1+2\sum_{j\geq 1}q^{j^{2}}$

is another of Ramanujan’s well–known theta functions [2, p. 6]. Note that

$\frac{f_{4p}^{k}}{f_{2p}^{2k}}$

is a function of $q^{p}$, which means that, in order to prove this theorem, we
simply need to consider whether $pn+r=j^{2}$ for some $r$ and $j$. Since $r$
is assumed to be a quadratic nonresidue modulo $p$, we know that there can be
no solutions to the equation $pn+r=j^{2}$. This proves our result. ∎

We close this paper with two sets of brief remarks.

  1. (1)

We first return to the generalized cubic partitions and make the following
elementary observation. For all $c>1$ which satisfy $c\equiv 1\pmod{p}$ for
$p=5,7$ or $11$, it is clear that $a_{c}(n)$ “inherits” the corresponding
congruence modulo $5$, $7$, or $11$ as Ramanujan’s (1.1). That is to say, for
all $j\geq 0$ and $n\geq 0,$

$\displaystyle a_{5j+1}(5n+4)$ $\displaystyle\equiv 0\pmod{5},$ $\displaystyle
a_{7j+1}(7n+5)$ $\displaystyle\equiv 0\pmod{7},\textrm{\ \ and}$
$\displaystyle a_{11j+1}(11n+6)$ $\displaystyle\equiv 0\pmod{11}.$

  2. (2)

As we mentioned above, our emphasis in this paper has been on congruences
modulo a prime $p$ satisfied by $a_{c}(n)$ or $\overline{a}_{c}(n)$ for
particular values of $c$. We encourage the interested reader to consider
identifying and proving additional congruences for these families of functions
modulo powers of a prime akin to Theorem 1.1.

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