Hiding all list items except the current and next element Only current and next element (cNum) is to be shown; in case it is the last element in the structure - the first element should be shown.

var childNum = $('#h').roundabout("getChildInFocus");
var cNum = (Number(childNum)) + 2;
if (cNum == 5) cNum = 1;

$("#h li").hide();
$("#h li:nth-child(" + cNum + ")").show();

if (cNum == 4) {
    $("#h li:nth-child(1)").show();
} else {
    $("#h li:nth-child(" + (Number(cNum) + 1).toString() + ")").show();
}
Here are a few tips.

1)

childNum is not needed since it's only used once.

Old Code:

var childNum = $('#h').roundabout("getChildInFocus");
var cNum = (Number(childNum)) + 2;


New Code: 

var cNum = $('#h').roundabout("getChildInFocus") + 2;


2)

In general, adding a object to a string results in a string.
And adding a object to a number results in a number.
Use parentheses to make the conversion clear.

Old Code:

$("#h li:nth-child(" + (Number(cNum) + 1).toString() + ")").show();


New Code:

$("#h li:nth-child(" + (1 + cNum) + ")").show(); 


3)

Cache commonly referenced elements.
Old Code:

$("#h li:nth-child(" + cNum + ")").show();


New Code:

var $lis = $("#h li");
$lis.find(":nth-child(" + cNum + ")").show();


4)

Combine operations on selectors.
Old Code:

$("#h li").hide();
$("#h li:nth-child(" + cNum + ")").show();


New Code:

$("#h li").hide().find(":nth-child(" + cNum + ")").show();


5)

Resetting the value of cNum to get rid of the else condition gives the following.

Old Code:

if (cNum == 4) {
    $("#h li:nth-child(1)").show();
} else {
    $("#h li:nth-child(" + (1+cNum) + ")").show();
}


New Code 1:

if (cNum == 4) {
    cNum = 0;
} 
$("#h li:nth-child(" + (1+cNum) + ")").show();


New Code 2:

if (cNum == 5) {
    cNum = 1;
} 
$("#h li:nth-child(" + cNum + ")").show();  


As you can see New Code 2 is the same from the new code in tip 4. 
So you can delete this redundant code.

Final Code:

var cNum = 2 + $('#h').roundabout("getChildInFocus");
if (cNum == 5){
    cNum = 1;
}
$("#h li").hide().find(":nth-child(" + cNum + ")").show();