How to use Select2 with JSON via Ajax request?

My Select2 3.4.5 is not working with JSON data.

Here is my input box on HTML:

<input class='form-control col-lg-5 itemSearch' type='text' placeholder='select item' />

…and my JavaScript

$(".itemSearch").select2({
    placeholder: "Search for an Item",
    minimumInputLength: 2,
    ajax: {
        url: "/api/productSearch",
        dataType: 'json',
        quietMillis: 100,
        data: function (term, page) {
            return {
                option: term
            };
        },
        results: function (data, page) {
            var more = (page * 10) < data.total;
            return {
                results: data.itemName,
                more: more
            };
        }
    },
    formatResult: function (data, term) {
        return data;
    },
    formatSelection: function (data) {
        return data;
    },
    dropdownCssClass: "bigdrop",
    escapeMarkup: function (m) {
        return m;
    }
});

I made an API with Laravel 4 which returns a value whenever I type anything on my text box.

Here's the result if I type "test" on my text box:

[{"itemName":"Test item no. 1","id":5},
{"itemName":"Test item no. 2","id":6},
{"itemName":"Test item no. 3","id":7},
{"itemName":"Test item no. 4","id":8},
{"itemName":"Test item no. 5","id":9},
{"itemName":"Test item no. 6","id":10},
{"itemName":"Test item no. 7","id":11}]

I can't add the result to my Select2 dropdown. I think formatSelection and formatResult are causing the problem because I don't know what parameter should be placed on it. I don't know where to get those parameters like container, object and query and the values it should be returning, or is my JSON response is wrong?

Here you have an example

$("#profiles-thread").select2({
    minimumInputLength: 2,
    tags: [],
    ajax: {
        url: URL,
        dataType: 'json',
        type: "GET",
        quietMillis: 50,
        data: function (term) {
            return {
                term: term
            };
        },
        results: function (data) {
            return {
                results: $.map(data, function (item) {
                    return {
                        text: item.completeName,
                        slug: item.slug,
                        id: item.id
                    }
                })
            };
        }
    }
});

It's quite easy

for select2 v4.0.0 slightly different

$(".itemSearch").select2({
    tags: true,
    multiple: true,
    tokenSeparators: [',', ' '],
    minimumInputLength: 2,
    minimumResultsForSearch: 10,
    ajax: {
        url: URL,
        dataType: "json",
        type: "GET",
        data: function (params) {

            var queryParameters = {
                term: params.term
            }
            return queryParameters;
        },
        processResults: function (data) {
            return {
                results: $.map(data, function (item) {
                    return {
                        text: item.tag_value,
                        id: item.tag_id
                    }
                })
            };
        }
    }
});

In Version 4.0.2 slightly different Just in processResults and in result :

    processResults: function (data) {
        return {
            results: $.map(data.items, function (item) {
                return {
                    text: item.tag_value,
                    id: item.tag_id
                }
            })
        };
    }

You must add data.items in result. items is Json name :

{
  "items": [
    {"id": 1,"name": "Tetris","full_name": "s9xie/hed"},
    {"id": 2,"name": "Tetrisf","full_name": "s9xie/hed"}
  ]
}

Here I give you my example which contain --> Country flag, City, State, Country.

Here is my output.

Attach these two Cdn js or links.

<link href="https://cdnjs.cloudflare.com/ajax/libs/select2/4.0.10/css/select2.min.css" rel="stylesheet" />
<script src="https://cdnjs.cloudflare.com/ajax/libs/select2/4.0.10/js/select2.min.js"></script>

js script

//for apend flag of country.

function formatState (state) {
    console.log(state);
    if (!state.id) {
      return state.text;
    }
    var baseUrl = "admin/images/flags";
    var $state = $(
      '<span><img src="'+baseUrl+ '/' + state.contryflage.toLowerCase() + '.png"  class="img-flag" /> ' +state.text+ '</span>'
    );
    return $state;
  };


$(function(){
    $("#itemSearch").select2({
    minimumInputLength: 2,
    templateResult: formatState, //this is for append country flag.
    ajax: {
        url: URL,
        dataType: 'json',
        type: "POST",
        data: function (term) {
            return {
                term: term
            };
        },
        processResults: function (data) {
            return {
                results: $.map(data, function (item) {
                    return {
                        text: item.name+', '+item.state.name+', '+item.state.coutry.name,
                        id: item.id,
                        contryflage:item.state.coutry.sortname
                    }
                })
            };
        }

    }
});

Expected JSON response.

[
   {
      "id":7570,
      "name":"Brussels",
      "state":{
         "name":"Brabant",
         "coutry":{
            "sortname":"BE",
            "name":"Belgium",

         }
      }
   },
   {
      "id":7575,
      "name":"Brussels",
      "state":{
         "name":"Brabant Wallon",
         "coutry":{
            "sortname":"BE",
            "name":"Belgium",

         }
      }
   },
   {
      "id":7578,
      "name":"Brussel",
      "state":{
         "name":"Brussel",
         "coutry":{
            "sortname":"BE",
            "name":"Belgium",

         }
      }
   },

]